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Their numerical coefficients may differ, as they do in $2x$ and $5x$."},{"text":"coefficient","correct":false,"explanation":"The coefficient is the numerical multiplier, which is $2$ in $2x$ and $5$ in $5x$. The shared feature that makes these like terms is their variable part."},{"text":"numerical part","correct":false,"explanation":"The numerical parts of $2x$ and $5x$ are different, so this cannot define like terms. Like terms instead have the same variable part."},{"text":"constant value","correct":false,"explanation":"Neither $2x$ nor $5x$ is a constant because each contains a variable. They are like terms because they share the same variable part, $x$."}]},{"id":"473f55a2-88ce-8573-037a-d8d0dda32209","cardType":"basic","points":2,"front":"Simplify $(4x-2)+(x+6)+(2x+1)$.","back":"$7x+5$","mcqOptions":[{"text":"$7x+5$","correct":true,"explanation":"Collecting the variable terms gives $4x+x+2x=7x$, while collecting the constants gives $-2+6+1=5$. Therefore the simplified expression is $7x+5$."},{"text":"$7x+9$","correct":false,"explanation":"The constant total becomes $9$ only if $-2$ is mistakenly treated as positive. Correctly calculating $-2+6+1=5$ gives the answer $7x+5$."},{"text":"$6x+5$","correct":false,"explanation":"This misses one of the seven lots of $x$, usually by failing to include the standalone $x$. The variable terms total $7x$, giving $7x+5$."},{"text":"$7x-5$","correct":false,"explanation":"The constants have been given the wrong overall sign. Since $-2+6+1=5$, and the variable terms total $7x$, the correct result is $7x+5$."}]},{"id":"8b073b77-1307-c074-1516-6757e1939414","cardType":"basic","points":1,"front":"In $3x-5$, what number term must be collected?","back":"$-5$","mcqOptions":[{"text":"$-5$","correct":true,"explanation":"The number term includes the sign immediately before it. In $3x-5$, the constant term is therefore negative five, written as $-5$."},{"text":"$5$","correct":false,"explanation":"Ignoring the subtraction sign changes the value of the constant. The sign belongs to the number term, so the term that must be collected is $-5$."},{"text":"$3$","correct":false,"explanation":"The $3$ is a coefficient because it multiplies $x$, rather than being a separate number term. The constant number term in the expression is $-5$."},{"text":"$-3$","correct":false,"explanation":"This incorrectly transfers the negative sign from the constant to the coefficient. The variable term is $3x$, while the number term that must be collected is $-5$."}]},{"id":"0b443eb1-4989-6d4f-ecb6-e16e22856ebb","cardType":"cloze","points":null,"front":"To form an equation from a known total, write an expression and set it [ ] to the given value.","back":"To form an equation from a known total, write an expression and set it equal to the given value.","mcqOptions":[{"text":"equal","correct":true,"explanation":"An equation states that two quantities have the same value. Therefore, an expression representing the unknown total must be set equal to the given total."},{"text":"greater than","correct":false,"explanation":"Using an inequality would state that the expression exceeds the known total rather than matches it. To form the required equation, set the expression equal to the given value."},{"text":"less than","correct":false,"explanation":"A less-than relationship forms an inequality and does not represent an exact known total. The expression must be set equal to the given value to form an equation."},{"text":"not equal","correct":false,"explanation":"A not-equal sign explicitly says the expression and known total differ, so it cannot model the stated total. The expression should instead be set equal to the value."}]},{"id":"90126d7f-72e5-a3e3-858e-003cc885a4b3","cardType":"cloze","points":null,"front":"Adding and [ ] are inverse operations; multiplying and [ ] are inverse operations.","back":"Adding and subtracting are inverse operations; multiplying and dividing are inverse operations.","mcqOptions":[{"text":"Adding and subtracting are inverse operations; multiplying and dividing are inverse operations.","correct":true,"explanation":"Subtracting reverses the effect of adding, while dividing reverses the effect of multiplying. Each pair can therefore be used to undo the other operation when solving equations."},{"text":"Adding and dividing are inverse operations; multiplying and subtracting are inverse operations.","correct":false,"explanation":"Dividing does not undo adding, and subtracting does not undo multiplying. The inverse of adding is subtracting, while the inverse of multiplying is dividing."},{"text":"Adding and multiplying are inverse operations; multiplying and dividing are inverse operations.","correct":false,"explanation":"Multiplication is repeated addition in some contexts, but it does not reverse addition. Subtracting is the inverse of adding, while dividing is the inverse of multiplying."},{"text":"Adding and subtracting are inverse operations; multiplying and adding are inverse operations.","correct":false,"explanation":"Adding cannot reverse multiplication because it changes a value by a fixed amount rather than a factor. Multiplication is undone by dividing, while adding is undone by subtracting."}]},{"id":"1c56e5d2-249d-2348-d903-957431a8c739","cardType":"basic","points":2,"front":"Rectangle length $3x+2$, width $x+4$: write the perimeter expression.","back":"$2(3x+2)+2(x+4)$","mcqOptions":[{"text":"$2(3x+2)+2(x+4)$","correct":true,"explanation":"A rectangle has two lengths and two widths. Doubling $3x+2$ and doubling $x+4$, then adding them, gives the perimeter expression $2(3x+2)+2(x+4)$."},{"text":"$(3x+2)(x+4)$","correct":false,"explanation":"Multiplying length by width calculates the rectangle’s area, not its perimeter. Perimeter requires two lengths and two widths, giving $2(3x+2)+2(x+4)$."},{"text":"$2(3x+2)+(x+4)$","correct":false,"explanation":"This includes two lengths but only one width, so one side is missing. A rectangle has two of each side, giving $2(3x+2)+2(x+4)$."},{"text":"$(3x+2)+2(x+4)$","correct":false,"explanation":"This includes two widths but only one length, so it does not cover all four sides. The complete perimeter is $2(3x+2)+2(x+4)$."}]},{"id":"e00dc03f-73ef-3885-5b35-189006f7cc92","cardType":"basic","points":2,"front":"Solve $8x+12=44$.","back":"$x=4$","mcqOptions":[{"text":"$x=4$","correct":true,"explanation":"Subtracting $12$ from both sides gives $8x=32$. Dividing both sides by $8$ then gives $x=4$, which satisfies the original equation."},{"text":"$x=7$","correct":false,"explanation":"This comes from adding $12$ to $44$ before dividing, rather than using the inverse operation. Subtracting $12$ gives $32$, so $x=4$."},{"text":"$x=5.5$","correct":false,"explanation":"Dividing $44$ by $8$ ignores the additional $12$ on the left-hand side. Subtract $12$ first, then divide $32$ by $8$ to obtain $x=4$."},{"text":"$x=32$","correct":false,"explanation":"Subtracting $12$ correctly produces $32$, but this stops before undoing the multiplication by $8$. Dividing $32$ by $8$ gives the solution $x=4$."}]},{"id":"9a6709a9-0f8c-aaa2-2ac7-8904bcd934fc","cardType":"cloze","points":null,"front":"Angles on a straight line add to [ ]; angles around a point add to [ ].","back":"Angles on a straight line add to 180°; angles around a point add to 360°.","mcqOptions":[{"text":"Angles on a straight line add to 180°; angles around a point add to 360°.","correct":true,"explanation":"A straight line is a half-turn, so its angles total $180°$. A full turn around a single point is twice as large, so those angles total $360°$."},{"text":"Angles on a straight line add to 90°; angles around a point add to 360°.","correct":false,"explanation":"A total of $90°$ describes a right angle, not a straight line. Angles on a straight line total $180°$, while angles around a point total $360°$."},{"text":"Angles on a straight line add to 180°; angles around a point add to 180°.","correct":false,"explanation":"A total of $180°$ represents only a half-turn, so it is insufficient around a point. The correct totals are $180°$ on a straight line and $360°$ around a point."},{"text":"Angles on a straight line add to 360°; angles around a point add to 180°.","correct":false,"explanation":"These totals reverse a full turn and a half-turn. A straight line is a $180°$ half-turn, whereas angles around a point make a $360°$ full turn."}]},{"id":"b1dc1b01-3093-99d6-ff61-8d0941939bbe","cardType":"cloze","points":null,"front":"A right angle is [ ]; angles in a triangle add to [ ].","back":"A right angle is 90°; angles in a triangle add to 180°.","mcqOptions":[{"text":"A right angle is 90°; angles in a triangle add to 180°.","correct":true,"explanation":"A quarter turn measures 90°, while the interior angles of every triangle have a total of 180°. These facts can be used to form equations involving unknown angles."},{"text":"A right angle is 180°; angles in a triangle add to 360°.","correct":false,"explanation":"This confuses a right angle with a straight angle and a triangle with a full turn. A right angle is 90°, and angles in a triangle total 180°."},{"text":"A right angle is 90°; angles in a triangle add to 360°.","correct":false,"explanation":"Although a right angle is correctly identified as 90°, 360° is the angle around a point. The interior angles of a triangle add to 180°."},{"text":"A right angle is 45°; angles in a triangle add to 180°.","correct":false,"explanation":"The triangle angle total is correctly given as 180°, but 45° is half a right angle. A right angle measures 90°."}]},{"id":"44cd4bed-f95c-344b-9ae0-ddcdd030a2e0","cardType":"basic","points":2,"front":"Solve $4x+(2x+6)=90$.","back":"$x=14$","mcqOptions":[{"text":"$x=14$","correct":true,"explanation":"Expanding and collecting terms gives $6x+6=90$. Subtracting 6 gives $6x=84$, and dividing both sides by 6 gives $x=14$."},{"text":"$x=16$","correct":false,"explanation":"This can result from adding 6 instead of subtracting it. The equation simplifies to $6x+6=90$, so $6x=84$ and $x=14$."},{"text":"$x=21$","correct":false,"explanation":"This comes from overlooking the additional $2x$ and dividing 84 by 4. Collecting both variable terms gives $6x=84$, so $x=14$."},{"text":"$x=15$","correct":false,"explanation":"This results from dividing 90 by 6 before removing the constant 6. First obtain $6x=84$, then divide by 6 to get $x=14$."}]},{"id":"123b7320-4e0f-8b84-e2c7-7c5e643d0f71","cardType":"basic","points":2,"front":"Solve $(2x+5)+(x+20)+(x-1)=180$.","back":"$x=39$","mcqOptions":[{"text":"$x=39$","correct":true,"explanation":"Collecting like terms gives $4x+24=180$. Subtracting 24 produces $4x=156$, and dividing by 4 gives $x=39$."},{"text":"$x=51$","correct":false,"explanation":"This comes from adding the constant total of 24 to 180 instead of subtracting it. The correct rearrangement is $4x=156$, giving $x=39$."},{"text":"$x=45$","correct":false,"explanation":"This comes from dividing 180 by 4 while ignoring the constant terms. The constants total 24, so $4x=156$ and therefore $x=39$."},{"text":"$x=41$","correct":false,"explanation":"This can result from combining $5+20-1$ incorrectly as 16. The constants actually total 24, giving $4x+24=180$ and hence $x=39$."}]},{"id":"bd486132-fb0e-e3b6-051f-4e4b9b3ddcab","cardType":"cloze","points":null,"front":"In a right-angled triangle, the two other angles add to [ ], not 180° by themselves.","back":"In a right-angled triangle, the two other angles add to 90°, not 180° by themselves.","mcqOptions":[{"text":"90°","correct":true,"explanation":"All three angles in a triangle total 180°. Since the right angle already accounts for 90°, the two remaining angles must together account for the other 90°."},{"text":"180°","correct":false,"explanation":"This is the total for all three angles, including the right angle. Subtracting the 90° right angle from 180° leaves 90° for the other two angles."},{"text":"360°","correct":false,"explanation":"This confuses the angle total of a triangle with the angle around a point. A triangle totals 180°, leaving 90° after its right angle is removed."},{"text":"45°","correct":false,"explanation":"Each remaining angle would be 45° only in an isosceles right-angled triangle. In every right-angled triangle, the two remaining angles together add to 90°."}]},{"id":"be697e9b-c8bd-93f9-168c-67382aed7c93","cardType":"basic","points":1,"front":"What expression gives the area of a triangle?","back":"1/2 × base × height","mcqOptions":[{"text":"1/2 × base × height","correct":true,"explanation":"A triangle with a given base and perpendicular height has half the area of the corresponding rectangle or parallelogram, so its area is 1/2 × base × height."},{"text":"base × height","correct":false,"explanation":"This calculates the area of a rectangle or parallelogram rather than a triangle. A triangle occupies half that area, so the correct expression is 1/2 × base × height."},{"text":"2 × base × height","correct":false,"explanation":"This doubles the corresponding rectangle’s area instead of halving it. A triangle’s area is found using 1/2 × base × height."},{"text":"1/2 × base + height","correct":false,"explanation":"This incorrectly adds the height rather than multiplying all three factors. The area of a triangle is calculated using 1/2 × base × height."}]},{"id":"a0257991-76e4-a0be-cbed-62baacb2faa8","cardType":"basic","points":1,"front":"How is perpendicular height measured relative to the base?","back":"At 90° to the base.","mcqOptions":[{"text":"At 90° to the base.","correct":true,"explanation":"Perpendicular lines meet at a right angle, so the perpendicular height must be measured at 90° to the chosen base, using the shortest distance from the opposite vertex."},{"text":"At 45° to the base.","correct":false,"explanation":"A 45° line is diagonal rather than perpendicular. The height used in the triangle area formula must meet the base at 90°."},{"text":"At 180° to the base.","correct":false,"explanation":"An angle of 180° forms a straight line along the base rather than a height. Perpendicular height is measured at 90° to the base."},{"text":"At 60° to the base.","correct":false,"explanation":"A 60° line may be a sloping side, but it is not perpendicular to the base. The required height must meet the base at 90°."}]},{"id":"425b425f-9147-a878-780c-9d594aff45d4","cardType":"basic","points":2,"front":"Right-angled triangle sides $5x$ and $4x$, area 360: what equation is formed?","back":"$1/2\\times 5x\\times 4x=360$","mcqOptions":[{"text":"$1/2\\times 5x\\times 4x=360$","correct":true,"explanation":"The two perpendicular sides act as the base and height. Substituting $5x$ and $4x$ into the triangle area formula gives $1/2\\times 5x\\times 4x=360$."},{"text":"$5x\\times 4x=360$","correct":false,"explanation":"This uses the rectangle area formula and omits the factor of one-half. For a triangle, the equation must be $1/2\\times 5x\\times 4x=360$."},{"text":"$1/2\\times (5x+4x)=360$","correct":false,"explanation":"This adds the two side lengths instead of multiplying the base and perpendicular height. The triangle area equation is $1/2\\times 5x\\times 4x=360$."},{"text":"$2\\times 5x\\times 4x=360$","correct":false,"explanation":"This doubles the base-height product rather than halving it. Since a triangle is half the corresponding rectangle, the required factor is $1/2$."}]},{"id":"22ee776e-5402-457c-7fb3-813cba719b98","cardType":"basic","points":2,"front":"Why use $x=6$, not $x=-6$, when $x^2=36$ and $x$ is a length?","back":"A length cannot be negative.","mcqOptions":[{"text":"A length cannot be negative.","correct":true,"explanation":"Solving $x^2=36$ algebraically gives $x=6$ or $x=-6$, but a geometric length must be non-negative, so only $x=6$ is valid."},{"text":"A length must be an even number.","correct":false,"explanation":"Lengths do not have to be even; they can be odd, fractional or irrational. The negative solution is rejected because a length cannot be negative."},{"text":"A square root is always positive.","correct":false,"explanation":"The equation $x^2=36$ has two algebraic solutions, $6$ and $-6$, because both square to 36. Only $6$ is used because a length cannot be negative."},{"text":"A negative number cannot be squared.","correct":false,"explanation":"Negative numbers can be squared, and $(-6)^2=36$. The value $-6$ is excluded only because $x$ represents a length, which cannot be negative."}]},{"id":"da3bb39f-8197-d848-acc4-0d8ba5ea8406","cardType":"basic","points":1,"front":"Parallelogram opposite sides $4x+1$ and $2x+13$: what equation is formed?","back":"$4x+1=2x+13$","mcqOptions":[{"text":"$4x+1=2x+13$","correct":true,"explanation":"Opposite sides of a parallelogram are equal in length, so the expressions representing those sides must be set equal: $4x+1=2x+13$."},{"text":"$4x+1+2x+13=0$","correct":false,"explanation":"Adding both side expressions and equating the result to zero does not represent equal opposite sides. The required equation is $4x+1=2x+13$."},{"text":"$4x+1=2(x+13)$","correct":false,"explanation":"The second side is already represented by $2x+13$; the 13 is not multiplied by 2. Equal opposite sides give $4x+1=2x+13$."},{"text":"$4x+2=2x+13$","correct":false,"explanation":"This changes the constant in the first given expression from 1 to 2. The expressions must be copied accurately, giving $4x+1=2x+13$."}]},{"id":"6ac95a46-486c-3737-c34a-ba47edd03b4b","cardType":"basic","points":1,"front":"What expression gives the area of a parallelogram?","back":"base × perpendicular height","mcqOptions":[{"text":"base × perpendicular height","correct":true,"explanation":"A parallelogram can be rearranged into a rectangle with the same base and perpendicular height, so its area is base × perpendicular height."},{"text":"base × sloping side length","correct":false,"explanation":"The sloping side is not generally perpendicular to the base, so using it gives the wrong area. The correct expression is base × perpendicular height."},{"text":"base + perpendicular height","correct":false,"explanation":"Adding lengths does not calculate area and produces a quantity measured in length units. The area of a parallelogram is base × perpendicular height."},{"text":"base × perpendicular height ÷ 2","correct":false,"explanation":"Dividing by 2 confuses a parallelogram with a triangle. A parallelogram occupies twice the area of the corresponding triangle, so use base × perpendicular height."}]},{"id":"8d14a5b0-3b3b-affc-3147-679957c93748","cardType":"basic","points":1,"front":"Sam has $x$ counters, Tia has twice as many, Ravi has 4 more than Tia. What is Ravi’s expression?","back":"$2x+4$","mcqOptions":[{"text":"$2x+4$","correct":true,"explanation":"Tia has twice Sam’s $x$ counters, giving $2x$. Ravi has 4 more than Tia, so his total is represented by $2x+4$."},{"text":"$2(x+4)$","correct":false,"explanation":"This doubles both Sam’s counters and the extra 4, incorrectly giving Ravi 8 additional counters. Only Tia’s amount is doubled, so Ravi has $2x+4$."},{"text":"$x+6$","correct":false,"explanation":"Twice as many counters means multiplying $x$ by 2, not adding 2. Adding Ravi’s extra 4 then gives the correct expression $2x+4$."},{"text":"$2x-4$","correct":false,"explanation":"Subtracting 4 would mean Ravi has 4 fewer counters than Tia. Ravi has 4 more, so the correct expression is $2x+4$."}]},{"id":"371fd856-4930-be5f-bf4e-575400b6f011","cardType":"basic","points":1,"front":"In the counters problem, $x=9$ is Sam’s counters. How many counters does Ravi have?","back":"22","mcqOptions":[{"text":"22","correct":true,"explanation":"Ravi’s number of counters is $2x+4$. Substituting $x=9$ gives $2(9)+4=18+4=22$ counters."},{"text":"26","correct":false,"explanation":"This results from incorrectly doubling the extra 4 as well as Sam’s 9 counters. Ravi has $2(9)+4=22$ counters."},{"text":"14","correct":false,"explanation":"This comes from adding only 5 to Sam’s amount and ignores that Tia has twice as many. Ravi has $2(9)+4=22$ counters."},{"text":"18","correct":false,"explanation":"This calculates Tia’s number by doubling Sam’s 9 counters but forgets Ravi’s additional 4. Ravi therefore has $18+4=22$ counters."}]},{"id":"ab7429ba-5b4d-50de-a905-b6c6278044c9","cardType":"basic","points":1,"front":"Alex is $x$, Ben is $3x$, Cara is 5 younger than Ben. What is Cara’s expression?","back":"$3x-5$","mcqOptions":[{"text":"$3x-5$","correct":true,"explanation":"Ben’s age is three times Alex’s, so it is $3x$. Cara is 5 years younger than Ben, giving the expression $3x-5$."},{"text":"$3(x-5)$","correct":false,"explanation":"This subtracts 5 from Alex’s age before tripling, making Cara 15 years younger than Ben. Subtract 5 from Ben’s age instead, giving $3x-5$."},{"text":"$3x+5$","correct":false,"explanation":"Adding 5 would make Cara 5 years older than Ben. Since she is 5 younger, her age is represented by $3x-5$."},{"text":"$x-5$","correct":false,"explanation":"This subtracts 5 from Alex’s age rather than Ben’s. Ben is $3x$, so Cara’s age is 5 less than this: $3x-5$."}]},{"id":"f350aad3-bb98-182c-3c93-63d267b602ae","cardType":"basic","points":2,"front":"Simplify the ratio $15:45:40$.","back":"$3:9:8$","mcqOptions":[{"text":"$3:9:8$","correct":true,"explanation":"The greatest common factor of 15, 45 and 40 is 5. Dividing every part by 5 gives the simplified ratio $3:9:8$."},{"text":"$1:3:8$","correct":false,"explanation":"This divides 15 and 45 by 15 but divides 40 by 5, changing the ratio. Every part must be divided by 5, giving $3:9:8$."},{"text":"$3:9:40$","correct":false,"explanation":"This divides the first two parts by 5 but leaves the final part unchanged. All three parts must be divided equally, producing $3:9:8$."},{"text":"$5:15:8$","correct":false,"explanation":"This divides the first two parts by 3 but the final part by 5, so the proportions change. Dividing every part by 5 gives $3:9:8$."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Component 1"},"heading":"Forming and Solving Equations","title":"Forming and Solving Equations Flashcards","topbar":{"backHref":"/gcse/maths/eduqas/flashcards"},"topicSidebar":"$L39"}]}],"topicId":"3ccde8cd-96ba-4a48-8438-4e75d08d4491","walled":"$L3a"}],"$L3b","$L3c"]