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Linear graphs gradients intercepts and parallel lines - Mathematics Year 10

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Introduction

Gradient describes how the vertical value changes for each horizontal unit. The intercept is a value at a specified axis, not another name for any point on the line.

Find four line equations and interpret gradient/intercept in two contexts. Use two-point differences, substitution and actual point checks; distinguish parallel, coincident, horizontal and vertical lines.

Find four line equations and verify their points

For a nonvertical straight line, y=mx+c expresses the gradient m and y-intercept c. At x=0, the product mx is 0, so the output is c. Between any two different horizontal coordinates, the gradient is $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}, x_{2} \neq x_{1} .$ Subtract coordinates in the same order in numerator and denominator.

For the source line through(0, 5) with gradient −2, the intercept is 5 immediately. Therefore y=−2 x+5. Moving one unit right reduces y by 2, so(1, 3) and(2, 1) belong as well. A negative gradient does not mean every output is negative; the starting value and the input determine the actual sign.

Four complete line equations. Cover the gradient, intercept and equation columns. Find m first, substitute a known point to determine c, then check another supplied point or a constructed new point.

Task	Given information	Gradient m	Y-intercept c	Equation
1	Through(0, 5), gradient −2	−2	5	y=−2 x+5
2	Parallel to y=3 x −4, through(2, 7)	3	1	y=3 x+1
3	Through(1, 4) and(5, 12)	2	2	y=2 x+2
4	Through(−2, 7) and(4, −2)	−3/2	4	y=−1.5 x+4

For row 2, a parallel line to y=3 x −4 has the same gradient 3. Use the point(2, 7): 7=3×2+c gives c=1. The answer is y=3 x+1. It is distinct from the original because the intercepts 1 and −4 differ. A second point(3, 10) follows by increasing x by 1 and y by 3.

For row 3, the changes are 12−4=8 vertically and 5−1=4 horizontally. Thus m=2. Substitute(1, 4) into 4=2×1+c to obtain c=2. Check(5, 12) in y=2 x+2. Reversing both subtractions gives(4−12)/(1−5)=−8/−4=2, the same gradient; reversing only one would incorrectly change its sign.

For row 4, the changes are −2−7=−9 and 4−(−2)=6, giving m=−9/6=−3/2. Substituting(−2, 7) gives 7=3+c and c=4. Both points satisfy y=−1.5 x+4: at −2 the output is 7, and at 4 it is −2. Keep the exact fraction −3/2 or terminating decimal −1.5; these representations are equal.

An intercept need not equal a point’s y coordinate when that point has nonzero x. In row 3, the given y value 4 at x 1 is not the intercept. Substitution determines the output at x 0, and independently shows the line passes through(0, 2).

Plot, compare parallels and distinguish intercepts

Distinct nonvertical straight lines are parallel when their gradients match and their intercepts differ. If both m and c match, they are the same line rather than two distinct parallels. A horizontal line y=c has gradient 0. A vertical line x=k has no finite gradient because its horizontal change is 0; it cannot be represented by y=mx+c with a finite m.

The source y=−2 x+5 has the following sample outputs. The original point plot uses evenly spaced integer y rows and equally spaced x positions. It marks sampled points rather than replacing the continuous straight line by disconnected measurements.

x	y=−2 x+5
0	5
1	3
2	1
3	−1

 y
 5 ●
 4 |
 3 |   ●
 2 |
 1 |       ●
 0 +----------------→ x
-1 |           ●
   0   1   2   3

Text alternative: points(0, 5), (1, 3), (2, 1), (3, −1) have horizontal change 1 and vertical change −2. The y-intercept is(0, 5). The x-intercept solves 0=−2 x+5 and is x=2.5, a different kind of intercept. Both intercept meanings should be named explicitly.

For the source parallel pair, y=3 x −4 andy=3 x+1, the output difference is always 5 at the same input. Trying to find an intersection gives 3 x −4=3 x+1 and hence −4=1, which is impossible. If both equations werey=3 x+1, all points of that line would be shared. Equal gradients alone do not distinguish coincidence from distinct parallelism.

The actual original Corbettmaths tutorial calculates a gradient from two coordinate points. Track vertical change divided by horizontal change and keep the subtraction order consistent. Its given points are separate from our four tasks; verify our equations with their own supplied coordinates. Original: corbettmaths, Gradient between two points - Corbettmaths. The written explanations and native tasks work without sound or the external player.

A first-error check catches the reciprocal mistake. For row 3, the vertical change 8 and horizontal change 4 give 2, not 4/8=.5. A candidate gradient.5 would move only 2 vertical units across the four-unit horizontal gap, so it cannot connect y 4 to y 12. Explain the geometric mismatch as well as correcting the fraction.

Interpret two rate contexts and transfer methods

A gradient is a rate whose units come from output units divided by input units. An intercept is an output at input 0 and inherits the output’s units. Numerical values without units can hide the difference between a per-item cost, a depth-loss rate and a geometric slope.

Two complete interpretations. These are stated mathematical models. The first input is a nonnegative whole count; the second is a nonnegative time while the predicted depth remains nonnegative.

Task	Model and units	Gradient meaning	Intercept meaning	Checked prediction
1	C=4+2.5 n; C in pounds, n items	£2.50 peritem	£4 fixed charge	At n 6, C£19
2	h=80−3 t; h in cm, t in minutes	−3 cm/min	80 cm initial depth	At t 10, h 50 cm

For the invented cost model C=4+2.5 n, each extra item adds£2.50 and the fixed charge is£4. At 6 items, the amount 4+15=£19. The total is not directly proportional to n because the nonzero intercept means doubling n does not double C. A second supplier model C=6+2.5 n has the same per-item rate but stays£2 higher at each n; the graph lines are distinct parallels.

For the depth model h=80−3 t, the initial depth is 80 cm and the constant rate is a loss of 3 cm each minute. After 10 minutes, the prediction is 50 cm. The model reaches 0 at t=80/3 minutes, or 26 minutes 40 seconds. Extend the algebra beyond that input and it predicts negative depth, which is outside this physical interpretation. Thus use 0≤t≤80/3 for the stated emptying model unless a new phase is specified.

For unfamiliar transfer, a line has gradient 4 and passes through(3, 5). Substitution 5=4×3+c gives c=−7, so y=4 x −7. At x 4, the output 9 is 4 above the supplied y 5, checking the gradient. A line parallel to y=−.5 x+2 through(4, 1) has c=3 and equationy=−.5 x+3; the two intercepts differ by 1.

For a boundary case, two points with the same x coordinate, for example(2, 1) and(2, 6), determine the vertical line x=2. The gradient fraction would divide 5 by 0 and is undefined, not 0. A horizontal line through(1, 4) and(5, 4) instead has vertical change 0 and nonzero horizontal change, so it is y=4 with gradient 0.

Keep four derived equations, two unit-aware context interpretations, point substitutions and an unfamiliar transfer. State whether a line is nonvertical before using gradient-intercept form. A sample-point plot or a numerical model is not evidence of an actual measuring experiment; later retrieval needs a real new response.

Interactive Tasks

Quiz: Test Your Knowledge

Source gradient −2 through 0, 5. Equation? (y=−2 x+5) (!y=5 x −2) (!y=2 x+5) (!y=−2 x −5)

Parallel to y=3 x −4 through 2, 7. Intercept? (1) (!-4) (!7) (!3)

Through 1, 4 and 5, 12. Gradient? (2) (!0.5) (!8) (!4)

Same two points. Equation? (y=2 x+2) (!y=2 x+4) (!y=0.5 x+3.5) (!y=4 x)

Through −2, 7 and 4, −2. Gradient? (-1.5) (!1.5) (!-2/3) (!4)

Source y=−2 x+5. X-intercept? (2.5) (!5) (!-2) (!0)

Same gradient and different intercepts imply distinct nonvertical lines are? (Parallel) (!Coincident) (!Always perpendicular) (!Both vertical)

Model C=4+2.5 n. Meaning of 4? (Fixed charge in pounds) (!Pounds per extra item) (!Number of items) (!Gradient)

Model h=80−3 t. Depth at 10 minutes? (50 cm) (!77 cm) (!30 cm) (!110 cm)

Line through 2, 1 and 2, 6. Gradient? (Undefined because horizontal change is 0) (!0) (!5) (!2)

Memory Game

Gradient	Vertical change divided by horizontal change
Y-intercept	Output where the horizontal input is 0
X-intercept	Input where the vertical output is 0
Parallel	Distinct straight lines with matching direction
Coincident	Representing the same line
Domain	Inputs permitted by the formula and context

Drag and Drop

y=−2 x+5	Source given-gradient line
y=3 x+1	Source parallel line through 2, 7
y=2 x+2	Two-point line through 1, 4 and 5, 12
y=−1.5 x+4	Two-point negative-gradient line
x=2	Vertical line through the two equal-x points

Crossword Puzzle

GRADIENT	Vertical rate per horizontal unit
INTERCEPT	Crossing of a named coordinate axis
PARALLEL	Distinct straight lines sharing direction
COINCIDENT	Describes two representations of the same line
VERTICAL	Having constant x coordinate
HORIZONTAL	Having constant y coordinate

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Read the source intercept: Explain why the given point at x 0 identifies c.
Read a negative gradient: Find the output change for one step right.
Distinguish axes: Compare y-intercept 5 with x-intercept 2.5.
Interpret units: Read pounds per item and centimetres per minute.

Standard

Find four line equations: Derive m and c and check actual supplied points.
Interpret two contexts: State rate, initial value, domain and prediction.
Explain parallelism: Separate matching gradient from matching entire equation.
Repair a reciprocal-gradient error: Use the point changes 8 and 4 and a geometric check.

Advanced

Transfer to new data: Find y=4 x −7 from gradient 4 and point 3, 5.
Transfer a negative parallel line: Find y=−.5 x+3 through 4, 1.
Analyse a physical cutoff: Derive 26 minutes 40 seconds before the depth model becomes invalid.
Handle vertical and horizontal cases: Explain undefined gradient versus zero gradient.

Learning Check

Four equations: Use gradient and point substitution.
Source point plot and intercepts: Name both axes explicitly.
Parallel and exceptional lines: Distinguish geometry cases.
Two contexts: State units, domains and checked predictions.
New-value transfer: Check the new points and rates.
Error and evidence: Keep methods and contextual limits.

Worked checks and assessment guidance

y=−2 x+5; y=3 x+1; y=2 x+2; y=−1.5 x+4. Gradients −2, 3, 2, −3/2 and intercepts 5, 1, 2, 4 reproduce the supplied data. Reversing both point differences preserves the gradient; reversing only one changes it incorrectly.
At x 0, 1, 2, 3 the outputs are 5, 3, 1, −1. Y-intercept is 0, 5; the x-intercept is 2.5, 0. A negative gradient does not require every y value to be negative.
Same finite gradient and different intercepts give distinct parallels; the same entire equation gives coincidence. x=2 is vertical with undefined gradient; y=4 is horizontal with gradient 0. The equal-x point pair would require division by 0.
C=4+2.5 n has£2.50 peritem and£4 fixed charge; at 6 items it gives£19. h=80−3 t has −3 cm/min and 80 cm initial depth; at 10 minutes it gives 50 cm. Nonnegative depth restricts t to 0 through 80/3 minutes.
Gradient 4 through 3, 5 givesy=4 x −7, and at 4 the output 9 confirms the one-step change 4. A parallel to y=−.5 x+2 through 4, 1 has equationy=−.5 x+3. Emptying 80 cm at 3 cm/min takes 80/3 minutes=26 minutes 40 seconds.
Gradient uses rise/run, not run/rise. Intercepts are values at a named axis, not arbitrary point coordinates. Keep four equations, two context interpretations, actual point checks and independent transfer; neither a model nor later intended practice proves an observed physical performance.

Learning evidence

Keep four derived line equations, two context interpretations with rate units and domains, actual point checks, exceptional vertical/horizontal cases and new transfer.

OER and Wikipedia

Department for Education:GCSE Mathematics subject content and assessment objectives. Source area:GCSE Mathematics subject content > Scope of study > Algebra - Graphs items 8 to 10. Year10 allocation is editorial within the GCSE pathway;Higher-only extensions are explicitly labelled. Original examples,teaching and feedback;Commons images retain their actual attribution,and videos identify original publishers. Reviewed12November2026.

2D Cartesian Coordinates.svg, H Padleckas, Public domain. Unchanged original; caption and learning activities are original.

Translation illustration.svg, Oleg Alexandrov, Public domain. Unchanged original; caption and learning activities are original.