1.2.3 Simultaneous equations

What you'll learn

What it means to solve two equations simultaneously.
How to solve two linear equations using elimination and substitution.
How to solve a system containing one linear and one quadratic equation.
How to check solutions and interpret them as intersections of graphs.

What are simultaneous equations?

A pair of simultaneous equations consists of two equations involving the same variables. You must find values of the variables that satisfy both equations at the same time.

For example,

x+y=7,x−y=1\begin{aligned} x+y&=7,\\ x-y&=1 \end{aligned}

has the solution $x = 4$ and $y = 3$ , because these values make both equations true.

Definition

Solution of simultaneous equations

A solution is an ordered pair $(x, y)$ that satisfies every equation in the system.

The graphical meaning

Each equation in two variables represents a graph. A simultaneous solution is therefore a point where the graphs intersect.

Two distinct straight lines usually meet once, giving one solution.
Parallel straight lines do not meet, giving no solutions.
Identical straight lines have infinitely many points in common.
A straight line and a quadratic curve can meet at zero, one or two points.

The graph below shows the line $y = x + 1$ intersecting the parabola $y=x^2-1$ at $(- 1, 0)$ and $(2, 3)$ .

The line y = x + 1 and parabola y = x squared minus 1 intersecting at negative 1 comma 0 and 2 comma 3

Key Idea

Equations as intersections

Solving simultaneous equations algebraically means finding the coordinates of every intersection of their graphs.

Prerequisite: rearranging equations

Before using substitution, you need to be able to make one variable the subject of an equation. The subject is the variable written alone on one side.

For example, rearranging $3 x + 2 y = 11$ to make $y$ the subject gives

y=11−3x2.y=\frac{11-3x}{2}.

Whatever operation you apply to one side of an equation must also be applied to the other side.

Example

Making a variable the subject

Make $x$ the subject of $4 x - 3 y = 10$ .

Add $3 y$ to both sides to isolate the term containing $x$ :
$4 x = 10 + 3 y .$
Divide every term by 4:
$x=10+3y4.x=\frac{10+3y}{4}.$

Solving linear equations by elimination

A linear equation contains variables only to the first power. Its graph is a straight line.

The elimination method combines the equations so that one variable disappears. You can add or subtract equations, provided the eliminated variable has equal or opposite coefficients.

Definition

Coefficient

A coefficient is the numerical factor multiplying a variable. In $5 x - 2 y = 9$ , the coefficients of $x$ and $y$ are 5 and $- 2$ respectively.

When the coefficients already match

If one variable has equal coefficients with the same sign, subtract the equations. If it has opposite coefficients, add the equations.

Example

Eliminating a variable directly

Solve

3x+2y=16,x+2y=8.\begin{aligned} 3x+2y&=16,\\ x+2y&=8. \end{aligned}

The coefficients of $y$ are equal, so subtract the second equation from the first:
$(3 x + 2 y) - (x + 2 y) = 16 - 8,$
giving $2 x = 8$ , so $x = 4$ .
Substitute $x = 4$ into one of the original equations:
$4 + 2 y = 8.$
Hence $2 y = 4$ , so $y = 2$ .
Check both equations:
$3 (4) + 2 (2) = 16$
and
$4 + 2 (2) = 8.$
Therefore the solution is $(4, 2)$ .

When the coefficients do not match

You may multiply one or both equations by a constant to create matching coefficients. You must multiply every term in the equation.

Example

Creating equal coefficients

Solve

2x+3y=12,5x−2y=11.\begin{aligned} 2x+3y&=12,\\ 5x-2y&=11. \end{aligned}

The lowest convenient matching coefficients of $y$ are 6 and $- 6$ . Multiply the first equation by 2 and the second by 3:
$4x+6y=24,15x−6y=33.\begin{aligned} 4x+6y&=24,\\ 15x-6y&=33. \end{aligned}$
Add the new equations. The $y$ terms cancel:
$19 x = 57,$
so $x = 3$ .
Substitute $x = 3$ into $2 x + 3 y = 12$ :
$6 + 3 y = 12,$
so $y = 2$ . The solution is $(3, 2)$ .

Common Mistake

Multiplying only part of an equation

When scaling an equation, multiply every term on both sides. For example, multiplying $2 x + 3 y = 12$ by 2 gives $4 x + 6 y = 24$ , not $4 x + 3 y = 24$ .

Tip

Choose the easier variable

Look for coefficients that are already equal, opposite, or have a small common multiple. This reduces arithmetic and the chance of an error.

Solving linear equations by substitution

The substitution method involves expressing one variable in terms of the other, then replacing that variable in the second equation.

It is particularly efficient when a variable is already isolated, such as $y = 2 x - 5$ , or has coefficient 1 or $- 1$ .

Example

Substituting a linear expression

Solve

y=3x−4,2x+y=11.\begin{aligned} y&=3x-4,\\ 2x+y&=11. \end{aligned}

Replace $y$ in the second equation with $3 x - 4$ :
$2 x + (3 x - 4) = 11.$
Solve the resulting equation in $x$ :
$\Rightarrow 5x=15 \Rightarrow x=3.$
Use $y = 3 x - 4$ to find the other variable:
$y = 3 (3) - 4 = 5.$
Therefore the solution is $(3, 5)$ .

Common Mistake

Missing brackets during substitution

If you substitute a multi-term expression into an equation, put it in brackets first. For example, replacing $y$ with $3 x - 4$ in $2 y$ gives $2 (3 x - 4)$ , so both terms are multiplied by 2.

Choosing between elimination and substitution

For two linear equations, either method can produce the same solution.

Use elimination when coefficients match easily.
Use substitution when one variable is already isolated or can be isolated simply.
If fractions appear immediately under one method, consider whether the other will be cleaner.

Key Idea

Method choice

The best method is usually the one that keeps the algebra simple. You are not required to use the same method for every pair of equations.

One linear and one quadratic equation

A quadratic equation contains a squared variable, such as $x^2$ . When one equation is linear and the other is quadratic, substitution is normally the most direct method.

The standard process is:

Rearrange the linear equation to make one variable the subject.
Substitute that expression into the quadratic equation.
Solve the resulting quadratic.
Substitute each possible value back to find the matching value of the other variable.

Because a line can intersect a quadratic curve twice, you must usually expect up to two solutions.

Example

Finding two line–curve intersections

Solve

y=x+1,y=x2−1.\begin{aligned} y&=x+1,\\ y&=x^2-1. \end{aligned}

Both expressions equal $y$ , so set them equal:
$x+1=x^2-1.$
Rearrange into quadratic form:
$x^2-x-2=0.$
Factorise:
$(x - 2) (x + 1) = 0.$
Therefore $x = 2$ or $x = - 1$ .
Substitute each value into $y = x + 1$ :
$\Rightarrow y=3,$
and
$\Rightarrow y=0.$
The two simultaneous solutions are $(2, 3)$ and $(- 1, 0)$ . These are the two intersections shown on the graph.

Substitution involving brackets

If the linear equation gives an expression such as $y = 5 - x$ , then substituting into $y^2$ produces $5-x)^2$ . The brackets are essential.

Example

Substituting into a squared variable

Solve

x+y=5,x+y2=11.\begin{aligned} x+y&=5,\\ x+y^2&=11. \end{aligned}

Rearrange the linear equation to obtain $x = 5 - y$ , then substitute into the second equation:
$5-y)+y^2=11.$
Rearrange and factorise:
$y2−y−6=0⇒(y−3)(y+2)=0.y^2-y-6=0 \Rightarrow (y-3)(y+2)=0.$
Hence $y = 3$ or $y = - 2$ .
Use $x = 5 - y$ for each value:
$\Rightarrow x=2,$
and
$\Rightarrow x=7.$
The solutions are $(2, 3)$ and $(7, - 2)$ .

Common Mistake

Discarding the second quadratic root

A quadratic may give two valid values. Substitute both roots back into the linear equation and report both ordered pairs unless the question imposes a restriction.

How many solutions are possible?

After substitution, the resulting quadratic may have two, one or no real roots. This tells you how many times the line meets the quadratic curve.

For $ax^2+bx+c=0$ , the discriminant is $b^2-4ac$ .

If $b^2-4ac>0$ , there are two distinct real solutions.
If $b^2-4ac=0$ , there is one repeated real solution; the line is tangent to the curve.
If $b^2-4ac<0$ , there are no real solutions; the graphs do not meet.

Definition

Tangent

A tangent is a line that touches a curve at a point without crossing it locally. For a line and a quadratic, this corresponds to a repeated root.

Example

Showing that no real intersection exists

Determine whether $y = x + 1$ and $y=x^2+x+3$ have any real simultaneous solutions.

Set the two expressions for $y$ equal:
$x+1=x^2+x+3.$
Rearrange:
$x^2+2=0,$
so $x^2=-2$ .
No real number has a negative square, so there are no real solutions. Equivalently, the quadratic has negative discriminant, confirming that the graphs do not intersect.

Exam technique

In the exam

Label equations or align them carefully so that signs and coefficients are easy to track.
Choose elimination for convenient matching coefficients and substitution when a variable is already isolated.
For a linear–quadratic system, solve the entire quadratic and find the corresponding second coordinate for every root.
Check each ordered pair in both original equations, especially after squaring, expanding brackets or handling negative values.

Self review

Check yourself

Can you solve two linear equations whose coefficients must first be multiplied?
Why can a line and a quadratic curve have two simultaneous solutions?
How would you recognise algebraically that two graphs have no real point of intersection?

What you'll learn

What it means to solve two equations simultaneously.
How to solve two linear equations using elimination and substitution.
How to solve a system containing one linear and one quadratic equation.
How to check solutions and interpret them as intersections of graphs.

What are simultaneous equations?

A pair of simultaneous equations consists of two equations involving the same variables. You must find values of the variables that satisfy both equations at the same time.

For example,

x+y=7,x−y=1\begin{aligned} x+y&=7,\\ x-y&=1 \end{aligned}

has the solution $x = 4$ and $y = 3$ , because these values make both equations true.

Definition

Solution of simultaneous equations

A solution is an ordered pair $(x, y)$ that satisfies every equation in the system.