Solving a system of equations means finding values that satisfy two equations in two unknowns at the same time.
If you buy 10 pieces of fruit, apples and pears together, and pay 13 dollars, one equation for the count and one for the cost must be solved together before the numbers of apples and pears are fixed.
There are two routes: elimination, which adds or subtracts the equations to wipe out one unknown, and substitution, which rewrites one equation as y = … and puts it into the other.
On a graph each equation is a straight line, and the coordinates of the point where the two lines meet are exactly the solution that satisfies both.
A point that meets both conditions
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Two lines whose slopes and y-intercepts change, and the coordinates of their intersection
1
1
-1
3
👀 Where two lines meet
①Every point on line 1 satisfies y = ax + b.
②Every point on line 2 satisfies y = cx + d.
③Only the intersection lies on both lines, so it alone satisfies both equations; its coordinates are the solution of the system. Moving the lines in the picture shows those coordinates exactly, fractions included.
Elimination: wiping out one unknown
📝 Solving by elimination
①Let x + y = 5 … (1) and x − y = 1 … (2).
②Adding (1) and (2) side by side removes y: 2x = 6, so x = 3.
③Putting x = 3 into (1) gives 3 + y = 5, so y = 2.
④Check: in (2), 3 − 2 = 1 also holds.
Key idea of elimination
make the coefficients of one unknown equal (or opposite), then subtract (or add) the equations
If the coefficients differ, first multiply each equation by a suitable number.
Substitution: putting one equation into the other
📝 Solving by substitution
①Let y = 2x − 1 … (1) and 3x + y = 9 … (2).
②Putting the y from (1) into (2) gives 3x + (2x − 1) = 9.
③Simplifying, 5x − 1 = 9 and 5x = 10, so x = 2.
④Putting x = 2 into (1) gives y = 2 × 2 − 1 = 3.
Key idea of substitution
rewrite one equation as y = (expression in x) → substitute it for y in the other
Substitution is convenient when an unknown has coefficient 1 or an equation is already in the form y = ….
Number of solutions and the graph
💡 How many solutions?
①If the slopes differ, the lines meet at one point and there is one solution.
②If the slopes are equal and the y-intercepts differ, the lines are parallel and never meet, so there is no solution.
③If both the slopes and the y-intercepts are equal, the lines coincide and there are infinitely many solutions. Setting the two slopes equal in the picture shows both cases.
Worked examples
Example 1
Solve the system x + y = 5, x − y = 1 by elimination.
1
Adding the two equations side by side removes y.
(x + y) + (x − y) = 5 + 1 ⇒ 2x = 6
2
Substitute x = 3 into one equation to find y.
x = 3, y = 5 − 3 = 2
▸ x = 3, y = 2
Elimination matches the coefficients of one unknown and adds or subtracts to remove it.
Example 2
Solve the system y = 2x − 1, 3x + y = 9 by substitution.
1
Put the y of the first equation in place of y in the second.
3x + (2x − 1) = 9 ⇒ 5x = 10
2
Substitute x = 2 into the first equation.
x = 2, y = 2·2 − 1 = 3
▸ x = 2, y = 3
Substitution is convenient when one equation is already y = (expression).
Exam checklist
Strategy for solving systems
elimination: match coefficients → add or subtract substitution: y = … → substitute
Both methods come down to removing one unknown and leaving a single linear equation.
Grade-8 school exam type
The system x + ay = 7, 2x − y = 4 has solution x = 3, y = 2. What is the constant a?
①1
②2
③3
④4
⑤5
▸ ② 2
1
Substitute the solution x = 3, y = 2 into the first equation.
3 + a·2 = 7
2
Solve for a.
2a = 4 ⇒ a = 2
🎯 Exam points
①Use elimination when the coefficients are simple and substitution when an equation is already y = ….
②When subtracting equations, every term of the subtracted equation changes sign.
③After solving, check the answer in both equations.
④In word problems with two unknowns, write two equations from two conditions.
⑤On a graph, the coordinates of the intersection of the two lines are the solution.