Unit 3 · Linear RelationshipsF2 · Slope & Lines · Guidance

Equations of Lines & Linear Models

Build line equations, interpret their parameters, and recognize parallel and perpendicular lines.

29 min
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7 steps · about 29 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

Intercept: A coordinate where a line crosses an axis.
Parallel lines: Coplanar lines with equal slopes that never meet.
Perpendicular lines: Lines meeting at 90 degrees with negative-reciprocal slopes.

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Try it first

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Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.

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The big idea

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See the structure

A line is determined by a slope and one point. Point-slope form keeps that information visible; slope-intercept form makes the intercept visible.

Key idea

Name quantities before calculating

Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.

Build a reliable method

Parallel nonvertical lines have equal slopes. Perpendicular nonvertical lines have slopes whose product is $-1$ .

Explore in Desmos

Recover an intercept from a point

The line must have slope 3 and pass through

(3,13)

. Read

b=4

, then set the slider to 4 and confirm that the line passes exactly through the point.

The calculator is live: change anything and see what happens.

Recover an intercept from a point

Degree mode

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Stretch to a 36

In a model $y=mx+b$ , the intercept is the predicted starting value. It is meaningful only when $x=0$ belongs to the context.

Trap

A believable answer to the wrong question

The choices often include a correct intermediate number. Before selecting it, reread the highlighted ask and check the unit, sign, and domain.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Find an intercept from a point and slope

Problem

What is the

y

-intercept of a line with slope 3 through

(2,11)

Substitute into

y=mx+b

11=3(2)+b

Which slope-intercept relationship uses the given point?

b=11-6=5

Which substitution confirms the line has the required slope and point?

Worked example

Example 2: Build a parallel line through a new point

Problem

Which equation represents the line parallel to

y=-2x+7

through

(4,1)

Parallel lines keep slope

-2

1=-2(4)+b

Which slope-intercept relationship uses the given point?

b=9

, so

y=-2x+9

Which substitution confirms the line has the required slope and point?

Faded example · you finish it

Example 3: Complete the setup and result

Problem

A perpendicular line to slope

3/4

has what slope?

Choose the setup that follows the stated relationship.

Choose the resulting value.

Desmos way

Check perpendicular slopes by their product

Assign the original slope with m=3/4 and its negative reciprocal with n=-4/3.

Type m*n and read

-1

. A product of

-1

confirms the slopes are perpendicular; changing only the sign or only taking a reciprocal would fail this check.

Check perpendicular slopes by their product

Degree mode

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.