Simultaneous equationsIB MYP Maths Extended: Revision notes

Section 1

What simultaneous equations are

Simultaneous equations are two or more equations with the same unknowns, which must both be true at the same time. For two unknowns $x$ and $y$ we need two equations. The solution is the pair of values $(x,y)$ that satisfies both. For example, $y=2x+1$ and $x+y=7$ are both true when $x=2$ and $y=5$ . There are three methods: substitution, elimination and graphs.

Key termssimultaneous equationssolution

Section 2

Substitution

Use substitution when one equation already gives a variable in terms of the other, such as $y=2x+1$ . Replace that variable in the second equation. Example: $y=2x+1$ and $x+y=7$ . Substitute: $x+(2x+1)=7$ , so $3x=6$ and $x=2$ . Then $y=2(2)+1=5$ . If neither equation is in that form, rearrange one of them first, e.g. $3x+y=11$ becomes $y=11-3x$ .

Key termssubstitution

Common mistake

Forgetting brackets when substituting. Replacing $y$ by $x+1$ in $5-2y$ must give $5-2(x+1)$ , not $5-2x+1$ .

Section 3

Elimination

Use elimination when both equations are in the form $ax+by=c$ . Make the coefficients of one variable equal in size, then add or subtract to remove it. Example: $3x+2y=16$ and $5x-2y=8$ . The $y$ terms are $+2y$ and $-2y$ , so add: $8x=24$ , $x=3$ . Substitute: $9+2y=16$ , so $y=3.5$ . If no variable matches, multiply one or both equations first. For $2x+3y=13.5$ and $3x+y=11.5$ , multiply the second by $3$ to get $9x+3y=34.5$ , then subtract: $7x=21$ . Rule: same signs, subtract; different signs, add.

Key termseliminationcoefficient

Common mistake

Subtracting only some terms. When you subtract equations, subtract every term, including the right-hand sides.

Exam tip

Substitute your answers into the other original equation to check them.

Section 4

Solving graphically

Each linear equation is a straight line. The solution of the pair is the point of intersection, where the lines cross. Draw both lines (by hand or with graphing technology) and read off the coordinates. For $y=2x+1$ and $x+y=7$ the lines cross at $(2,5)$ , so $x=2$ and $y=5$ . If the lines are parallel they never cross, so there is no solution. The graphical method is quick, but readings may only be approximate.

Key termspoint of intersectionparallel

Section 5

Setting up equations from word problems

Choose letters for the two unknowns and say what they mean.
Write one equation for each piece of information.
Solve by substitution or elimination.
Answer the question in words with units, and check in the original problem. Example: 2 adult and 1 child ticket cost 30; 1 adult and 2 child cost 24. Then $2a+c=30$ and $a+2c=24$ , giving $a=12$ and $c=6$ . Check: $2(12)+6=30$ and $12+2(6)=24$ .

Key termsunknowndefine variables

Exam tip

Finish by answering what was asked. Finding $x$ and $y$ is not always the final answer.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Simultaneous equations

Two lines have equations $y=2x+1$ and $x+y=7$ .
Explain what the solution of the pair of equations tells you about the graphs of the two lines.2 marks
A pair of simultaneous equations is $3x+2y=16$ and $5x-2y=8$ .
Find the value of $y$ , and check that your solution satisfies the other equation.2 marks
At a bookshop in Singapore, 2 notebooks and 3 pens cost 13.50 Singapore dollars (SGD). Also, 3 notebooks and 1 pen cost 11.50 SGD.
Let $n$ be the price of a notebook and $p$ the price of a pen, in SGD. Write down two equations and eliminate one variable to obtain an equation in a single variable.3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).