Simultaneous Equations

Spec reference: Algebra - Solving equations and inequalities
Key idea: Solve two simultaneous equations to find values of two unknowns.

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What are simultaneous equations?

Two equations with two unknowns. We find the values of both unknowns that satisfy both equations at the same time.

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Method 1: Elimination

Add or subtract the equations to eliminate one variable.

Example

Solve: $3x+2y=13$ and $x+2y=7$

Subtract equation 2 from equation 1:

$(3x+2y)-(x+2y)=13-7$

$2x=6\Rightarrow x=3$

Substitute into equation 2: $3+2y=7\Rightarrow y=2$

$$\boxed{{\displaystyle x=3,y=2}}$$

Example

Solve: $2x+3y=11$ and $4x-y=9$

Multiply equation 2 by 3: $12x-3y=27$

Add to equation 1: $14x=38\Rightarrow x=\frac{19}{7}$...

Alternatively, multiply equation 1 by 2: $4x+6y=22$

Subtract equation 2: $7y=13\Rightarrow y=\frac{13}{7}$

$x=\frac{9+\frac{13}{7}}{4}$...

Better: multiply eq1 by 1, eq2 by 3 to eliminate $y$:

$2x+3y=11$
$12x-3y=27$

Add: $14x=38\Rightarrow x=\frac{19}{7}$

For cleaner numbers: $2x+3y=11$ and $4x-y=9$

Multiply eq2 by 3: $12x-3y=27$. Add to eq1: $14x=38$, $x=\frac{19}{7}$, $y=\frac{5}{7}$

$$\boxed{{\displaystyle x=\frac{19}{7},y=\frac{5}{7}}}$$

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Method 2: Substitution

Rearrange one equation and substitute into the other.

Example

Solve: $y=2x-1$ and $3x+y=14$

Substitute $y=2x-1$ into eq2:

$3x+(2x-1)=14$

$5x=15\Rightarrow x=3$

$y=2(3)-1=\boxed{{\displaystyle 5}}$

$$\boxed{{\displaystyle x=3,y=5}}$$

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Graphical method

The solution is where the two lines intersect.

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Exam tips

Watch out for

• Always check your answer in both original equations
• If eliminating, make sure the coefficients match before adding/subtracting
• Substitution is easiest when one equation is already in the form $y=\dots$

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Test Yourself

Question 1 of 5

Solve: $x + y = 7$ $x - y = 3$