GED Mathematical Reasoning
Slope & Linear Graphs
Slope is steepness — how much a line climbs for every step it takes to the right. A road sign showing 6% grade is a slope. On a graph it tells you the rate: dollars per hour, miles per gallon.
The words first
- slope
- rise over run — how far up divided by how far across
- y-intercept
- where the line crosses the vertical axis — the starting value
- rise / run
- the up-and-down change / the left-and-right change
- y = mx + b
- the standard form: m is the slope, b is the starting value
The method
THE FORMULA
rise y₂ − y₁
slope = —————— = —————————
run x₂ − x₁
Subtract in the SAME order on top and bottom.
y first minus y second, then x first minus x second.
READING THE SIGN
going up as you read right → positive slope
going down as you read right → negative slope
flat line → slope 0
y = mx + b
m = slope = the rate, the per-something
b = y-intercept = the starting amount, the flat fee
A gym costs $25 to join plus $10 a month:
y = 10x + 25 (10 per month, 25 to start)
Worked all the way through
The problem
Find the slope of the line through the points (2, 3) and (6, 11).
- Step 1 — Label the numbers so you do not mix them up.First point: x₁ = 2, y₁ = 3. Second point: x₂ = 6, y₂ = 11.
- Step 2 — Write the formula.slope = (y₂ − y₁) / (x₂ − x₁)
- Step 3 — Fill in the y values on top.11 − 3 = 8
- Step 4 — Fill in the x values on the bottom, in the SAME order.6 − 2 = 4
- Step 5 — Divide.8 / 4 = 2
On the board
(2, 3) and (6, 11)
the two points
slope = (y₂ − y₁) / (x₂ − x₁)
the formula
= (11 − 3) / (6 − 2)
same order top and bottom
= 8 / 4
= 2
Answer
slope = 2
A second one, in a different form
A line has the equation y = −3x + 8. What are its slope and y-intercept?
y = mx + b
the standard form
y = −3x + 8
match the pieces
m = −3
the slope — the number on x
b = 8
the y-intercept — the plain number
Answer
slope −3, y-intercept 8
The usual mistake
Flipping the slope fraction into run over rise.
Check yourself
The order has to match. If you do 11 − 3 on top you must do 6 − 2 on the bottom, not 2 − 6. Flipping one and not the other gives you the right number with the wrong sign.
Practice it
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