SME
Grade 8 Math · Problem type
Lines written ax + by = c
Use this for “Which point is on both 2x + y = 9 and x − y = 3?”
Chart
They cross at (4, 1). −2 × 4 + 9 = 1
1 × 4 − 3 = 1
This is an example. Type your own values to start a new problem.
a
x
+
b
y
=
c
d
x
+
(
e
)
y
=
f
m₁
First slope
worked out (not typed)
b₁
First intercept
worked out (not typed)
m₂
Second slope
worked out (not typed)
b₂
Second intercept
worked out (not typed)
x
Solution x
worked out (not typed)
y
Solution y
worked out (not typed)
Formulas
m₁
=
−
a
÷
b
(first
slope
=
−x
in
the
first
÷
y
in
the
first)
−2 = −2 ÷ 1
b₁
=
c
÷
b
(first
intercept
=
right
side
of
the
first
÷
y
in
the
first)
9 = 9 ÷ 1
m₂
=
−
d
÷
e
(second
slope
=
−x
in
the
second
÷
y
in
the
second)
1 = −1 ÷ (−1)
b₂
=
f
÷
e
(second
intercept
=
right
side
of
the
second
÷
y
in
the
second)
−3 = 3 ÷ (−1)
x
=
(
b₂
−
b₁
)
÷
(
m₁
−
m₂
)
(solution
x
=
(second
intercept
−
first
intercept)
÷
(first
slope
−
second
slope))
4 = (−3 − 9) ÷ (−2 − 1)
y
=
m₁
×
x
+
b₁
(solution
y
=
first
slope
×
solution
x
+
first
intercept)
1 = −2 × 4 + 9
y
=
m₂
×
x
+
b₂
(solution
y
=
second
slope
×
solution
x
+
second
intercept)
1 = 1 × 4 + (−3)
a
×
x
+
b
×
y
=
c
(x
in
the
first
×
solution
x
+
y
in
the
first
×
solution
y
=
right
side
of
the
first)
2 × 4 + 1 × 1 = 9
d
×
x
+
e
×
y
=
f
(x
in
the
second
×
solution
x
+
y
in
the
second
×
solution
y
=
right
side
of
the
second)
1 × 4 + (−1) × 1 = 3
Assumptions
•
Rewrite ax + by = c as y = −(a ÷ b)x + c ÷ b: the slope and the intercept.
•
Then find where the two lines cross, as for y = mx + b.
•
Both lines need y in them (b and e not 0): an upright line x = k is not y = mx + b.
Step-by-step
WE KNOW
a = 2 (x in the first), b = 1 (y in the first), c = 9 (right side of the first), d = 1 (x in the second), e = −1 (y in the second), f = 3 (right side of the second)
FIND
first slope (m₁), first intercept (b₁), second slope (m₂), second intercept (b₂), solution x (x), solution y (y)
Step 1 · Find first slope (m₁)
Use m₁ = −a ÷ b (first slope = −x in the first ÷ y in the first)
Take
ax
to
the
other
side
and
divide
by
b
:
the
slope.
m₁
=
−
2
1
m₁
=
−2
Step 2 · Find first intercept (b₁)
Use b₁ = c ÷ b (first intercept = right side of the first ÷ y in the first)
Divide
the
right
side
by
b
:
the
intercept.
b₁
=
9
1
b₁
=
9
Step 3 · Find second slope (m₂)
Use m₂ = −d ÷ e (second slope = −x in the second ÷ y in the second)
The same for the second line.
m₂
=
−
1
−1
m₂
=
1
Step 4 · Find second intercept (b₂)
Use b₂ = f ÷ e (second intercept = right side of the second ÷ y in the second)
The second line’s intercept.
b₂
=
3
−1
b₂
=
−3
Step 5 · 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
=
−3 − 9
−2 − 1
−
2
x
+
9
=
x
−
3
Take
x
from
both
sides:
−
3
x
+
9
=
−3
Take
9
from
both
sides:
−
3
x
=
−12
Divide
both
sides
by
−3:
x
=
4
x
=
4
Step 6 · 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
×
4
+
9
y
=
−8
+
9
y
=
1
Check
Put the numbers back in. Both sides should match.
−2 = −2 ÷ 1 ✓
9 = 9 ÷ 1 ✓
1 = −1 ÷ (−1) ✓
−3 = 3 ÷ (−1) ✓
4 = (−3 − 9) ÷ (−2 − 1) ✓
1 = −2 × 4 + 9 ✓
1 = 1 × 4 + (−3) ✓
2 × 4 + 1 × 1 = 9 ✓
1 × 4 + (−1) × 1 = 3 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Nothing to review first. This is a starting point.
Related lessons
Systems of two linear equations
Main lesson
When do two plans cost the same?
Use this for “Gym A: $150 to join and $20 a month. Gym B: $60 and $35. When do they cost the same?”
How many of each
Use this for “8 tickets cost $41. Adult tickets are $7 and child tickets $4. How many of each?”
How many solutions does the system have?
Use this for “Do y = 3x − 2 and y = 3x + 5 have a solution?”