SME
Grade 8 Math · Problem type
Lines written ax + by = c
Use this for “A snack mix uses nuts at $4 a pound and raisins at $2 a pound; $16 in all” or “(4, y) is on 3x + 4y = 24: find y.”
Chart
y = (−1/2)x + 4
Slope m = −1/2: down 1 for every 2 across
Intercept k = 4: it crosses the y-axis at (0, 4)
y = −1/2 × 2 + 4 = 3 when x = 2
This is an example. Type your own values to start a new problem.
a
x
+
b
y
=
c
x
x
example
y
y
worked out (not typed)
m
Slope
worked out (not typed)
k
y-intercept
worked out (not typed)
Formulas
y
=
(
c
−
a
×
x
)
÷
b
(y
=
(constant
−
coefficient
of
x
×
x)
÷
coefficient
of
y)
3 = (12 − 1.5 × 2) ÷ 3
m
=
−
a
÷
b
(slope
=
−coefficient
of
x
÷
coefficient
of
y)
−
1
2
=
−1.5
÷
3
k
=
c
÷
b
(y-intercept
=
constant
÷
coefficient
of
y)
4 = 12 ÷ 3
Assumptions
•
ax + by = c is a line too. Put in x and solve for y.
•
Rearranged, y = −(a ÷ b)x + c ÷ b: slope −a ÷ b, intercept c ÷ b.
•
Each point (x, y) on the line is a mix that makes c.
Step-by-step
WE KNOW
a = 1.5 (coefficient of x), b = 3 (coefficient of y), c = 12 (constant), x = 2 (x)
FIND
y (y), slope (m), y-intercept (k)
Step 1 · Find y (y)
Use y = (c − a × x) ÷ b (y = (constant − coefficient of x × x) ÷ coefficient of y)
Take
a
×
x
from
both
sides,
then
divide
by
b
.
y
=
12 − 1.5 × 2
3
3
y
=
12
−
1.5
×
2
3
y
=
9
y
=
9
3
y
=
3
Step 2 · Find slope (m)
Use m = −a ÷ b (slope = −coefficient of x ÷ coefficient of y)
The
slope
of
the
same
line:
−
a
b
.
m
=
−
1.5
3
m
=
−
1
2
Step 3 · Find y-intercept (k)
Use k = c ÷ b (y-intercept = constant ÷ coefficient of y)
Its
y
-intercept:
y
when
x
=
0.
k
=
12
3
k
=
4
Check
Put the numbers back in. Both sides should match.
3 = (12 − 1.5 × 2) ÷ 3 ✓
−
1
2
=
−1.5
÷
3
✓
4 = 12 ÷ 3 ✓
Topic check
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