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same time."},{"text":"Two or more equations that must be solved separately from each other.","correct":false,"explanation":"Solving each equation with unrelated values misses the defining requirement. Simultaneous equations are two or more equations that must be true at the same time."},{"text":"Two or more equations that must have exactly the same coefficients.","correct":false,"explanation":"Equations do not need identical coefficients to be simultaneous. They are simultaneous when the same variable values must make all the equations true at the same time."},{"text":"Two or more equations that must use completely different variables.","correct":false,"explanation":"Having different variables does not define simultaneous equations. The equations usually share variables and must all be satisfied by the same values at the same time."}]},{"id":"d78a9909-be2f-2277-4e3c-e23e1cfe4e98","cardType":"basic","points":1,"front":"What does the solution to a pair of simultaneous equations represent on a graph?","back":"The coordinates of the point where the two lines intersect.","mcqOptions":[{"text":"The coordinates of the point where the two lines intersect.","correct":true,"explanation":"A solution satisfies both equations, so its coordinates lie on both lines. Graphically, this is precisely the point at which the two lines intersect."},{"text":"The gradients of the two lines where they have the same steepness.","correct":false,"explanation":"The gradient describes a line’s steepness, not the shared solution. The solution is given by the coordinates of the point where the two lines intersect."},{"text":"The intercepts of the points where the two lines cross the axes.","correct":false,"explanation":"The intercepts show where individual lines cross an axis, which is not generally their shared solution. The solution is the coordinates of the lines’ intersection point."},{"text":"The coordinates of the point lying exactly between both lines.","correct":false,"explanation":"The midpoint between two lines need not lie on either line and does not satisfy both equations. The solution is the coordinates of their intersection point."}]},{"id":"c7b45622-0ad7-10aa-5e52-9fe02c203811","cardType":"basic","points":2,"front":"How do you verify if a pair of values (e.g., $x=4, y=1$) is the correct solution?","back":"Substitute the values into both original equations to see if they work.","mcqOptions":[{"text":"Substitute the values into both original equations to see if they work.","correct":true,"explanation":"A simultaneous solution must satisfy every original equation. Substituting both values into both equations confirms whether each resulting numerical statement is true."},{"text":"Substitute the values into one original equation to see if they work.","correct":false,"explanation":"Checking only one equation can accept values that fail the other equation. Substitute the values into both original equations and verify that both statements are true."},{"text":"Add the values together in both original equations to see if they work.","correct":false,"explanation":"Adding the values together does not test whether they satisfy either equation. Each value must be substituted for its variable in both original equations."},{"text":"Swap the values before using both original equations to see if they work.","correct":false,"explanation":"Swapping the values tests the ordered pair $(y,x)$ rather than the proposed pair $(x,y)$. Substitute each value for its correct variable in both equations."}]},{"id":"62dd9487-ce19-28a4-13b9-803760b4615a","cardType":"cloze","points":null,"front":"Elimination means adding or subtracting equations so that [ ].","back":"Elimination means adding or subtracting equations so that one variable disappears.","mcqOptions":[{"text":"one variable disappears","correct":true,"explanation":"Adding or subtracting suitably arranged equations cancels matching variable terms. This makes one variable disappear, leaving an equation in the remaining variable."},{"text":"both variables remain","correct":false,"explanation":"Elimination aims to remove a variable, not preserve both of them. The equations are added or subtracted so that one variable disappears."},{"text":"each variable is isolated","correct":false,"explanation":"Adding or subtracting equations does not generally isolate each variable immediately. Elimination first makes one variable disappear so the remaining variable can be found."},{"text":"one coefficient doubles","correct":false,"explanation":"A coefficient can double when like-signed terms are added, but that is not the aim of elimination. The operation should make one variable disappear."}]},{"id":"0a7bfa2a-175d-0b28-5316-04d5a46ba2cb","cardType":"basic","points":1,"front":"What is a coefficient in algebra?","back":"The number multiplying a variable (e.g., in $4x$, the coefficient is $4$).","mcqOptions":[{"text":"The number multiplying a variable (e.g., in $4x$, the coefficient is $4$).","correct":true,"explanation":"A coefficient is the numerical factor attached to a variable. For example, $4x$ means $4\\times x$, so its coefficient is $4$."},{"text":"The letter representing a number (e.g., in $4x$, the coefficient is $x$).","correct":false,"explanation":"The variable is the letter representing an unknown quantity, not the coefficient. In $4x$, the number $4$ multiplies $x$ and is therefore the coefficient."},{"text":"The power applied to a variable (e.g., in $4x$, the coefficient is $1$).","correct":false,"explanation":"An exponent indicates repeated multiplication and is written as a power, whereas a coefficient multiplies a variable. In $4x$, the coefficient is $4$."},{"text":"The constant without a variable (e.g., in $4x$, the coefficient is $x$).","correct":false,"explanation":"A constant term has no variable attached, whereas a coefficient is the number multiplying a variable. In $4x$, that multiplying number is $4$."}]},{"id":"57ffab68-9029-da2a-2702-7ae5463fc074","cardType":"basic","points":1,"front":"In the elimination method, when should you add the two equations?","back":"When the coefficients of a variable are the same size but have opposite signs (e.g., $+2y$ and $-2y$).","mcqOptions":[{"text":"When the coefficients of a variable are the same size but have opposite signs (e.g., $+2y$ and $-2y$).","correct":true,"explanation":"Equal-sized coefficients with opposite signs cancel when added, since terms such as $+2y$ and $-2y$ sum to zero. This eliminates that variable."},{"text":"When the coefficients of a variable are the same size and have the same sign (e.g., $+2y$ and $+2y$).","correct":false,"explanation":"Equal-sized coefficients with the same sign double when added rather than cancelling. In this case, subtract the equations to eliminate the variable."},{"text":"When the coefficients of a variable are different sizes but have opposite signs (e.g., $+2y$ and $-3y$).","correct":false,"explanation":"Opposite signs alone do not guarantee cancellation when the coefficient sizes differ. First scale an equation so the coefficients have the same size, then add."},{"text":"When the coefficients of a variable are different sizes and have the same sign (e.g., $+2y$ and $+3y$).","correct":false,"explanation":"Different-sized coefficients with the same sign do not cancel through addition. The equations must first be scaled to create equal-sized coefficients suitable for elimination."}]},{"id":"1b246d81-c76f-28b8-952a-73e9a1e621d3","cardType":"basic","points":1,"front":"In the elimination method, when should you subtract the two equations?","back":"When the coefficients of a variable are the same size and have the same sign (e.g., $+3y$ and $+3y$).","mcqOptions":[{"text":"When the coefficients of a variable are the same size and have the same sign (e.g., $+3y$ and $+3y$).","correct":true,"explanation":"Equal-sized coefficients with the same sign cancel when one equation is subtracted from the other, since terms such as $+3y-(+3y)$ equal zero."},{"text":"When the coefficients of a variable are the same size but have opposite signs (e.g., $+3y$ and $-3y$).","correct":false,"explanation":"Subtracting equal-sized terms with opposite signs makes their size double rather than cancelling them. Add the equations when the coefficients have opposite signs."},{"text":"When the coefficients of a variable are different sizes and have the same sign (e.g., $+3y$ and $+2y$).","correct":false,"explanation":"Same signs alone are insufficient when the coefficient sizes differ, because subtraction leaves a non-zero variable term. First scale an equation to make the sizes equal."},{"text":"When the coefficients of a variable are different sizes and have opposite signs (e.g., $+3y$ and $-2y$).","correct":false,"explanation":"Different-sized coefficients with opposite signs do not cancel through subtraction. Scale the equations to equalise their sizes, then add terms with opposite signs."}]},{"id":"8e3b7125-77ed-5f74-14f5-666c5c32e8f0","cardType":"cloze","points":null,"front":"If the coefficients of a variable have the [ ], you subtract the equations; if they have opposite signs, you add them.","back":"If the coefficients of a variable have the same sign, you subtract the equations; if they have opposite signs, you add them.","mcqOptions":[{"text":"same sign","correct":true,"explanation":"Coefficients with the same sign cancel through subtraction because one is removed from the other. Coefficients with opposite signs instead cancel through addition."},{"text":"same size","correct":false,"explanation":"Coefficient size alone does not determine the operation because signs also matter. Subtract equal coefficients with the same sign, but add equal-sized coefficients with opposite signs."},{"text":"opposite sign","correct":false,"explanation":"Coefficients with opposite signs should be added, not subtracted, so that they cancel. Subtraction is used when equal-sized coefficients have the same sign."},{"text":"same variable","correct":false,"explanation":"Both equations normally contain the same variable regardless of which operation is needed. Subtract when its equal-sized coefficients have the same sign, and add when they have opposite signs."}]},{"id":"162c81b1-bd35-7749-f59a-268b41cc3f27","cardType":"cloze","points":null,"front":"If the coefficients of a variable have the same sign, you subtract the equations; if they have [ ], you add them.","back":"If the coefficients of a variable have the same sign, you subtract the equations; if they have opposite signs, you add them.","mcqOptions":[{"text":"opposite signs","correct":true,"explanation":"Opposite signs cancel when the equations are added because one coefficient is positive and the other is negative."},{"text":"the same signs","correct":false,"explanation":"Coefficients with the same signs reinforce rather than cancel when added. The correct fill is opposite signs, because a positive and negative coefficient cancel through addition."},{"text":"positive signs","correct":false,"explanation":"Two positive coefficients become larger when added, so the variable is not eliminated. The correct fill is opposite signs, which cancel when the equations are added."},{"text":"negative signs","correct":false,"explanation":"Two negative coefficients combine rather than cancel when added. The correct fill is opposite signs, because equal positive and negative coefficients sum to zero."}]},{"id":"49701109-4faa-93c9-e073-6b5153167219","cardType":"basic","points":1,"front":"What is the common mistake when subtracting one equation from another?","back":"Forgetting to subtract every term on both the left and right sides of the equals sign.","mcqOptions":[{"text":"Forgetting to subtract every term on both the left and right sides of the equals sign.","correct":true,"explanation":"Subtracting equations means subtracting the entire second equation, so every term on both sides must have the subtraction applied to it."},{"text":"Forgetting to divide every term on both the left and right sides of the equals sign.","correct":false,"explanation":"Division is not the required operation when one equation is being subtracted. The common mistake is failing to subtract every term on both sides of the equals sign."},{"text":"Forgetting to multiply every term on both the left and right sides of the equals sign.","correct":false,"explanation":"Multiplication is used earlier only when coefficients need matching. During subtraction, the key mistake is forgetting to subtract every term on both sides of the equations."},{"text":"Forgetting to substitute every term on both the left and right sides of the equals sign.","correct":false,"explanation":"Substitution is a different solving method and is not applied term by term here. When subtracting equations, every term on both sides must be subtracted."}]},{"id":"1da4766a-54db-8fe5-5783-9ab63c0942af","cardType":"basic","points":2,"front":"What must be done if neither $x$ nor $y$ have matching coefficients in a pair of equations?","back":"Multiply one or both whole equations by a number to make the coefficients match.","mcqOptions":[{"text":"Multiply one or both whole equations by a number to make the coefficients match.","correct":true,"explanation":"Multiplying a whole equation preserves its equality while producing matching coefficients, allowing one variable to be eliminated by adding or subtracting the equations."},{"text":"Add a number to one or both equations to make the coefficients match.","correct":false,"explanation":"Adding a number does not scale the coefficients and may alter only selected terms. Instead, multiply one or both whole equations to create matching coefficients."},{"text":"Multiply only the variable terms by a number to make the coefficients match.","correct":false,"explanation":"Multiplying only variable terms changes the equation and destroys the equality. Every term in one or both whole equations must be multiplied by the chosen number."},{"text":"Swap the variables in one or both equations to make the coefficients match.","correct":false,"explanation":"Swapping variables changes what the equations represent and does not produce an equivalent system. Multiply one or both whole equations to make corresponding coefficients match."}]},{"id":"bda815f8-7261-d0fe-c0e0-7df55383aa84","cardType":"cloze","points":null,"front":"When multiplying an equation to match coefficients, [ ] in the equation must be multiplied by that number.","back":"When multiplying an equation to match coefficients, every term in the equation must be multiplied by that number.","mcqOptions":[{"text":"every term","correct":true,"explanation":"An equation remains balanced only if the multiplier is applied to every term on both sides, including variables, constants and any negative terms."},{"text":"only the variable term","correct":false,"explanation":"Multiplying only the variable term changes the equation rather than forming an equivalent one. The correct fill is every term, including constants on both sides."},{"text":"only the coefficient","correct":false,"explanation":"Changing only one coefficient does not preserve the equality of the original equation. The correct fill is every term, all multiplied by the same number."},{"text":"the term being eliminated","correct":false,"explanation":"Scaling only the term intended for elimination produces a non-equivalent equation. The correct fill is every term, because the whole equation must be multiplied consistently."}]},{"id":"e500a9b7-93ea-b7a4-a98d-e00e34d741e0","cardType":"basic","points":1,"front":"What is the standard form for arranging simultaneous equations before solving?","back":"$$ax + by = c$ (variables on the left, number on the right).","mcqOptions":[{"text":"$$ax + by = c$ (variables on the left, number on the right).","correct":true,"explanation":"Standard form places both variable terms together on the left-hand side and the constant alone on the right-hand side."},{"text":"$$c = ax + by$ (number on the left, variables on the right).","correct":false,"explanation":"Although algebraically equivalent, this reverses the required standard arrangement. Standard form is $ax + by = c$, with variables left and the constant right."},{"text":"$$ax = c - by$ ($x$ on the left, $y$ and number on the right).","correct":false,"explanation":"This leaves the variable terms on opposite sides of the equation. Standard form collects both variables on the left as $ax + by = c$."},{"text":"$$by = c - ax$ ($y$ on the left, $x$ and number on the right).","correct":false,"explanation":"This isolates one variable term instead of collecting both on the same side. Standard form is $ax + by = c$, with the constant on the right."}]},{"id":"6ea1589f-8e93-8b12-4be1-01cd826137f0","cardType":"basic","points":1,"front":"How should the equation $3x = 12 + y$ be rearranged into standard form?","back":"$$3x - y = 12$","mcqOptions":[{"text":"$$3x - y = 12$","correct":true,"explanation":"Subtracting $y$ from both sides moves it to the left with a negative sign, leaving the constant $12$ on the right."},{"text":"$$3x + y = 12$","correct":false,"explanation":"This keeps $y$ positive after moving it across the equals sign. Subtracting $y$ from both sides gives the correct form $3x - y = 12$."},{"text":"$$3x - y = -12$","correct":false,"explanation":"The sign of $12$ should not change because it has not crossed the equals sign. Only $y$ moves, giving $3x - y = 12$."},{"text":"$$3x + y = -12$","correct":false,"explanation":"This incorrectly changes the signs of both $y$ and $12$. Subtract only $y$ from both sides to obtain $3x - y = 12$."}]},{"id":"ecfa022b-bdb3-ed91-994d-6eaa7eed33c0","cardType":"cloze","points":null,"front":"If solving simultaneous equations results in a fractional answer, you should [ ] as not all answers are whole numbers.","back":"If solving simultaneous equations results in a fractional answer, you should keep going carefully as not all answers are whole numbers.","mcqOptions":[{"text":"keep going carefully","correct":true,"explanation":"Simultaneous equations can legitimately have fractional solutions, so continue using exact fraction arithmetic and check the final values in both original equations."},{"text":"round it immediately","correct":false,"explanation":"Rounding during the calculation loses accuracy and may prevent the solution satisfying both equations. The correct fill is keep going carefully, retaining the exact fraction."},{"text":"change it to zero","correct":false,"explanation":"A fractional value is not automatically zero and replacing it changes the solution. The correct fill is keep going carefully, because fractional answers can be valid."},{"text":"start again completely","correct":false,"explanation":"A fraction does not by itself show that an error has occurred. The correct fill is keep going carefully, while checking each algebraic step and sign."}]},{"id":"408008f3-e76e-e983-dc55-84945ad72761","cardType":"basic","points":2,"front":"After finding the value of one variable (e.g., $x=4$), how do you find the value of the second variable?","back":"Substitute the known value into one of the original equations and solve.","mcqOptions":[{"text":"Substitute the known value into one of the original equations and solve.","correct":true,"explanation":"Replacing the known variable in either original equation leaves an equation containing only the second variable, which can then be solved directly."},{"text":"Substitute the known value into the final one-variable equation and solve.","correct":false,"explanation":"The final one-variable equation only confirms the value already found and contains no second variable. Substitute into one original equation to calculate the other value."},{"text":"Add the known value to both of the original equations and solve.","correct":false,"explanation":"Adding the value changes the equations and does not replace the known variable. Substitute it into one original equation, leaving only the second variable unknown."},{"text":"Divide both original equations by the known value and solve.","correct":false,"explanation":"Dividing the equations does not use the fact that one variable has a known value. Substitute that value into one original equation and solve for the other variable."}]},{"id":"0efdcc80-b7f6-3604-2507-acded17e9acc","cardType":"basic","points":2,"front":"Two sandwiches ($s$) and three drinks ($d$) cost £14.70. One sandwich and two drinks cost £8.70. What are the two equations?","back":"

$2s + 3d = 14.70$
$s + 2d = 8.70$

","mcqOptions":[{"text":"

$2s + 3d = 14.70$
$s + 2d = 8.70$

","correct":true,"explanation":"The coefficients represent the numbers of sandwiches and drinks in each purchase, while £14.70 and £8.70 are the corresponding totals."},{"text":"

$2s + 3d = 8.70$
$s + 2d = 14.70$

","correct":false,"explanation":"This swaps the two purchase totals. The order information gives $2s + 3d = 14.70$ for the first purchase and $s + 2d = 8.70$ for the second."},{"text":"

$3s + 2d = 14.70$
$2s + d = 8.70$

","correct":false,"explanation":"This reverses the numbers of sandwiches and drinks in both purchases. The correct coefficients are two and three in the first equation, then one and two in the second."},{"text":"

$2s + 3d = 14.70$
$2s + d = 8.70$

","correct":false,"explanation":"The second purchase has one sandwich and two drinks, not two sandwiches and one drink. Its equation must therefore be $s + 2d = 8.70$."}]},{"id":"512c4e03-768f-2d98-0cc9-f7ffe3a4707d","cardType":"cloze","points":null,"front":"In word problems, it is vital not to [ ] of your variables halfway through the calculation.","back":"In word problems, it is vital not to swap the meanings of your variables halfway through the calculation.","mcqOptions":[{"text":"swap the meanings","correct":true,"explanation":"Each variable must represent the same quantity throughout the calculation. Swapping their meanings would make later equations and substitutions inconsistent with the original definitions."},{"text":"swap the coefficients","correct":false,"explanation":"Coefficients may legitimately change when equations are multiplied during elimination. The essential rule is not to swap the meanings assigned to the variables themselves."},{"text":"reverse the signs","correct":false,"explanation":"Signs can legitimately reverse when multiplying an entire equation by a negative number. The key consistency requirement is not to swap the meanings of the variables."},{"text":"simplify the coefficients","correct":false,"explanation":"Simplifying coefficients can be a valid algebraic step when applied correctly. The mistake warned against is swapping the meanings of the variables during the calculation."}]},{"id":"1ca9bd55-c975-2fad-12c3-58af9ace41c7","cardType":"basic","points":1,"front":"How do you perform a final check of your answers for $x$ and $y$?","back":"Substitute both values into both original equations to ensure both sides match.","mcqOptions":[{"text":"Substitute both values into both original equations to ensure both sides match.","correct":true,"explanation":"A simultaneous solution must satisfy both original equations. Substituting both values into each equation confirms that every left-hand side equals its corresponding right-hand side."},{"text":"Substitute both values into one original equation to ensure both sides match.","correct":false,"explanation":"Checking only one equation can miss a pair that satisfies that equation but not the other. Both values must be substituted into both original equations."},{"text":"Substitute one value into both original equations to ensure both sides match.","correct":false,"explanation":"Each original equation contains both unknowns, so checking only one value is incomplete. Substitute both values into both equations and verify that the two sides match."},{"text":"Substitute both values into the final equation only to ensure both sides match.","correct":false,"explanation":"A final derived equation may not independently check the original system. Both calculated values must be substituted into both original equations to verify the solution fully."}]},{"id":"8170ca4c-dd93-9b99-565f-4a8e9eaf9a7d","cardType":"cloze","points":null,"front":"A pair of values is only a solution if it works in [ ] equations.","back":"A pair of values is only a solution if it works in both equations.","mcqOptions":[{"text":"both","correct":true,"explanation":"Simultaneous equations require one pair of values that satisfies the two equations at the same time. Therefore, the pair must work in both equations."},{"text":"only one of the","correct":false,"explanation":"Satisfying just one equation does not establish a simultaneous solution because many other pairs may also satisfy it. The values must work in both equations."},{"text":"at least one of the","correct":false,"explanation":"Working in at least one equation is insufficient for a simultaneous solution. The defining requirement is that the same pair of values satisfies both equations."},{"text":"neither of the","correct":false,"explanation":"A pair that satisfies neither equation cannot be a solution to the system. A valid simultaneous solution must make both original equations true."}]},{"id":"0cceb08b-5edc-7409-7747-2603c1fa08d5","cardType":"cloze","points":null,"front":"In the standard form $ax+by=c$, the [ ] is placed on the right side of the equation.","back":"In the standard form $ax+by=c$, the constant (number) is placed on the right side of the equation.","mcqOptions":[{"text":"constant (number)","correct":true,"explanation":"In $ax+by=c$, the variable terms are collected on the left and the constant value $c$, which contains no variable, is placed on the right."},{"text":"coefficient of x","correct":false,"explanation":"The coefficient of $x$ is $a$, and it remains attached to $x$ on the left. The constant $c$ is placed alone on the right."},{"text":"coefficient of y","correct":false,"explanation":"The coefficient of $y$ is $b$, which forms the term $by$ on the left. It is the constant $c$ that belongs on the right."},{"text":"variable term","correct":false,"explanation":"The variable terms $ax$ and $by$ are placed on the left in standard form. The constant number $c$ is isolated on the right."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Unit 1"},"heading":"Simultaneous Equations","title":"Simultaneous Equations Flashcards","topbar":{"backHref":"/gcse/maths/wjec/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"aad5cf14-7399-eb72-4ffc-439c188d81df","walled":"$L39"}],"$L3a","$L3b"]