A physiotherapy course costs a £40 assessment fee plus £22 per session. Written as y = mx + c, what is the slope and what does it represent?

62, the cost of the first session including the fee

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October 2026 - Application of Slope Study Guide

Slope is rise over run: the change in y divided by the change in x. In a word problem it is always a rate — cost per item, distance per hour, dose per kilogram — and reading it as a rate is what turns an abstract formula into an answerable question.

Mathematics guide Numbers and algebra · 18 scored questions 6 min read

Two rising lines of different steepness with a highlighted horizontal run and vertical rise between points. — Slope compares vertical change with horizontal change. For the same run, the line with the greater rise is steeper.

The formula and what it means

Between two points, slope is (y₂ - y₁) / (x₂ - x₁). The order of the points does not matter as long as you keep it consistent: subtract the y-values in the same order as the x-values, or the sign comes out wrong.

Read it aloud as "for every one step across, how far up or down". A slope of 4 means y climbs 4 for each 1 that x advances. A slope of 1/3 means y climbs 1 for every 3 across.

Four kinds of slope

Slope	Looks like	Means
Positive	Rises left to right	y increases as x increases
Negative	Falls left to right	y decreases as x increases
Zero	A horizontal line	y never changes — no relationship
Undefined	A vertical line	x never changes; division by zero

Slope from two points

Worked example

Find the slope of the line through (2, 5) and (6, 17).

Label them: x₁ = 2, y₁ = 5, x₂ = 6, y₂ = 17.
Rise: 17 - 5 = 12.
Run: 6 - 2 = 4.
Divide: 12 ÷ 4 = 3.

The slope is 3 — y rises by 3 for every 1 that x increases. Reversing both subtractions gives -12 / -4, which is still 3; reversing only one gives −3, which is the error to watch for.

Slope as a rate, in words

Word problems hide slope inside ordinary phrasing. These are the ones that recur.

"£4 per additional mile" — slope 4, in pounds per mile.
"loses 2 kg a month" — slope −2, because the value falls.
"2 mg for every kilogram of body weight" — slope 2, in mg per kg.
"a flat fee of £30 plus £5 an hour" — slope 5, y-intercept 30, giving y = 5x + 30.

A steeper line is not a larger value; it is a faster change. Two lines can cross, with the steeper one starting lower and ending higher — which is exactly the situation graph questions about "when does option B become cheaper" are built on.

Slope-intercept form

In y = mx + c, m is the slope and c is the y-intercept, the value of y when x is zero. Given a graph, read the intercept where the line crosses the vertical axis, then count rise over run between two clear points to get the slope.

Given a word problem, the fixed amount is c and the per-unit amount is m. That single mapping answers most questions in this topic without any algebra at all.

Terms to know

Slope: The rate of change of a line: rise over run.
Rise: The change in the y-values between two points.
Run: The change in the x-values between two points.
Y-intercept: Where a line crosses the y-axis — the value of y when x is 0.
Rate of change: How much one quantity changes per unit of another. Slope, in words.
Linear relationship: A relationship with a constant rate of change, so its graph is a straight line.

What the TEAS asks most

The same core ideas appear in different wording. If you can answer these without stopping to think, you have what this section requires.

What is the slope of the line through these two points?

Rise over run, subtracting the coordinates in the same order both times.

What does the slope represent in this situation?

A rate, with units: pounds per hour, kilograms per month, miles per gallon.

Which graph shows the fastest increase?

The steepest positive slope, regardless of where each line starts.

What is the equation of this line?

y = mx + c: slope from rise over run, c from where it crosses the y-axis.

Key points

Slope is rise over run — the change in y over the change in x.
Keep the subtraction order consistent or the sign inverts.
Positive rises, negative falls, zero is horizontal, vertical is undefined.
In a word problem, slope is the per-unit rate and the intercept is the fixed amount.
Steepness is speed of change, not size of value.

Put this TEAS topic into practice

Build a set on application of slope and use the rationale on every question to reinforce what you just reviewed.

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