Simultaneous equations · Years 9–10
In Simultaneous equations (Years 9–10) you practise 4 skills: Is it a solution of both equations? · Solving by substitution and elimination · The solution on a graph: where the lines meet · Word problems with simultaneous equations. Simultaneous equations are two equations with two unknowns. Their solution is a pair of values that makes both equations true at the same time.
In substitution, use one of the equations to find what one unknown is equal to, and substitute that expression into the other equation. You are left with an equation with one unknown. In elimination, multiply an equation by a number so that one of the unknowns has equal or opposite coefficients in the two equations. Subtract or add the equations, and that unknown cancels out. On a graph, each equation is a straight line, and the solution is the point where the lines meet. Parallel lines have no point in common, so the equations have no solution. If the two equations describe the same line, they have infinitely many solutions.
Tip: At the end, substitute the solution into both equations. If both hold, the solution is right.
What you learn
Is it a solution of both equations?
For example: Is (−4, 5) a solution of the simultaneous equations? −2x + 3y = 23 −x − 3y = −11
- Yes
- No
Answer and solution
The answer: Yes
Substitute into the first equation: −2 × (−4) + 3 × 5 = 23 The equation holds. Substitute into the second equation: −(−4) − 3 × 5 = −11 The equation holds. Both equations hold, so the pair is a solution of the simultaneous equations. The answer is yes.
Solving by substitution and elimination
For example: Solve the simultaneous equations: y = −2x − 4 3x − y = −21
Answer and solution
The answer: (−5, 6)
Substitute −2x − 4 for y in the second equation: 3x − (−2x − 4) = −21 3x + 2x + 4 = −21 5x = −25 x = −25 ÷ 5 = −5 Substitute x = −5 into the first equation: y = −2 × (−5) − 4 = 6 The solution: (−5, 6)
The solution on a graph: where the lines meet
For example: Read the solution of the simultaneous equations from the graph: y = −2x − 2 y = x + 7
Answer and solution
The answer: (−3, 4)
The lines meet at (−3, 4) . Check in both equations: y = −2 × (−3) − 2 = 4 y = −3 + 7 = 4 The solution: (−3, 4)
Word problems with simultaneous equations
For example: The sum of two numbers is 34, and the difference between them is 2. What is the larger number?
Answer and solution
The answer: 18
Call the larger number x and the smaller number y . The equations: x + y = 34 x − y = 2 Add the two equations: 2x = 36 x = 18 The larger number is 18.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).