SME
Grade 8 Math · Expressions & Equations
Systems of two linear equations
Use this for “Where do y = 2x − 1 and y = −x + 5 meet?” by graphing or by substitution.
Chart
They cross at (2, 3).
2 × 2 − 1 = 3
−1 × 2 + 5 = 3
This is an example. Type your own values to start a new problem.
y
=
m₁
x
+
(
b₁
)
y
=
m₂
x
+
b₂
x
Solution x
worked out (not typed)
y
Solution y
worked out (not typed)
Formulas
x
=
(
b₂
−
b₁
)
÷
(
m₁
−
m₂
)
(solution
x
=
(second
intercept
−
first
intercept)
÷
(first
slope
−
second
slope))
2 = (5 − (−1)) ÷ (2 − (−1))
y
=
m₁
×
x
+
b₁
(solution
y
=
first
slope
×
solution
x
+
first
intercept)
3 = 2 × 2 + (−1)
y
=
m₂
×
x
+
b₂
(solution
y
=
second
slope
×
solution
x
+
second
intercept)
3 = −1 × 2 + 5
Assumptions
•
The solution is the point on both lines: where they cross.
•
Set the two expressions for y equal, then solve for x.
•
Lines with the same slope and different intercepts are parallel: no solution.
•
Check the point in both equations.
Step-by-step
WE KNOW
m₁ = 2 (first slope), b₁ = −1 (first intercept), m₂ = −1 (second slope), b₂ = 5 (second intercept)
FIND
solution x (x), solution y (y)
Step 1 · Find solution x (x)
Use x = (b₂ − b₁) ÷ (m₁ − m₂) (solution x = (second intercept − first intercept) ÷ (first slope − second slope))
Set
the
two
right
sides
equal,
gather
x
on
one
side,
then
divide.
x
=
5 − (−1)
2 − (−1)
2
x
−
1
=
−
x
+
5
Add
x
to
both
sides:
3
x
−
1
=
5
Add
1
to
both
sides:
3
x
=
6
Divide
both
sides
by
3:
x
=
2
x
=
2
Step 2 · Find solution y (y)
Use y = m₁ × x + b₁ (solution y = first slope × solution x + first intercept)
Put
x
into
the
first
equation.
y
=
2
×
2
+
(−1)
y
=
4
+
(−1)
y
=
3
Check
Put the numbers back in. Both sides should match.
2 = (5 − (−1)) ÷ (2 − (−1)) ✓
3 = 2 × 2 + (−1) ✓
3 = −1 × 2 + 5 ✓
Topic check
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