Coordinate Geometry Basics: Equations of Lines from Points and Slope

1) Reading Slope and y-Intercept in y = mx + b

The slope-intercept form y = mx + b connects an algebraic equation to two visible features of a line on the coordinate plane:

• b is the y-intercept: the point where the line crosses the y-axis. It is always the point (0, b).
• m is the slope: how much y changes for a 1-unit change in x. It tells the line’s direction and steepness.

Example A: Identify features from an equation

Given y = 2x - 3:

• y-intercept: b = -3, so the line crosses the y-axis at (0, -3).
• slope: m = 2, meaning for every +1 in x, y goes up by +2.

Example B: Compare slopes by looking at m

Consider these equations:

• y = 3x + 1 has slope 3 (rises quickly).
• y = (1/3)x + 1 has slope 1/3 (rises slowly).
• y = -2x + 1 has slope -2 (falls as x increases).

Notice that all three share the same y-intercept (0, 1), so they all cross the y-axis at the same point but tilt differently.

Graph examples (described)

• If b is large and positive, the line crosses the y-axis above the origin.
• If b is negative, it crosses below the origin.
• If m is positive, the line goes up from left to right; if negative, it goes down.

2) Finding a Line’s Equation

2a) From a point and a slope

If you know a point (x1, y1) on the line and the slope m, use point-slope form:

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y - y1 = m(x - x1)

Then simplify into slope-intercept form y = mx + b if needed.

Example C: Point and slope, then simplify

Find the equation of the line with slope m = -3 passing through (2, 5).

Step 1: Substitute into point-slope form

y - 5 = -3(x - 2)

Step 2: Distribute

y - 5 = -3x + 6

Step 3: Add 5 to both sides

y = -3x + 11

So the equation is y = -3x + 11.

2b) From two points

If you are given two points, you can build the equation in two stages:

• Find the slope m from the two points.
• Use point-slope form with either point, then simplify.

Example D: Two points to slope to equation

Find the equation of the line through (-1, 4) and (3, -2).

Step 1: Compute the slope

m = (y2 - y1)/(x2 - x1) = (-2 - 4)/(3 - (-1)) = (-6)/4 = -3/2

Step 2: Use point-slope form (choose one point)

Using (-1, 4):

y - 4 = (-3/2)(x - (-1))

y - 4 = (-3/2)(x + 1)

Step 3: Distribute

y - 4 = (-3/2)x - 3/2

Step 4: Add 4 to both sides

y = (-3/2)x - 3/2 + 4

Step 5: Combine constants

Write 4 as 8/2:

y = (-3/2)x + ( -3/2 + 8/2 ) = (-3/2)x + 5/2

So the equation is y = (-3/2)x + 5/2.

Alternative method: Solve for b using y = mx + b

Once you know m, you can substitute a point into y = mx + b to find b.

From Example D, m = -3/2. Use point (3, -2):

-2 = (-3/2)(3) + b

-2 = -9/2 + b

Add 9/2 to both sides:

b = -2 + 9/2 = -4/2 + 9/2 = 5/2

So y = (-3/2)x + 5/2.

3) Graphing Lines from Equations

To graph a line given in slope-intercept form y = mx + b, use a two-part plan: plot the y-intercept, then use the slope to find more points.

Method: Intercept then slope steps

• Step 1: Plot (0, b).
• Step 2: Rewrite the slope as a fraction m = rise/run.
• Step 3: From the intercept, move run units horizontally and rise units vertically to locate a second point.
• Step 4: Draw the line through the points and extend it.

Example E: Graph y = (2/3)x - 1

• y-intercept: b = -1 so plot (0, -1).
• slope: m = 2/3 means rise +2, run +3.
• From (0, -1), move right 3 to x = 3, then up 2 to y = 1. Second point: (3, 1).
• Draw the line through (0, -1) and (3, 1).

Example F: Graph y = -4x + 2

• y-intercept: plot (0, 2).
• slope: -4 = -4/1. From (0, 2), move right 1 and down 4 to get (1, -2).
• Draw the line through (0, 2) and (1, -2).

When the equation is not already y = mx + b

If the equation is in another form, rearrange it to solve for y first.

Example G: Rearrange then graph

Graph 2y = 6x - 8.

Step 1: Solve for y

y = 3x - 4

Step 2: Plot y-intercept (0, -4)

Step 3: Use slope m = 3 = 3/1: from (0, -4) go right 1, up 3 to (1, -1).

4) Check Understanding: Match Graphs, Tables, and Equations

In coordinate geometry, the same line can be represented in multiple ways. Practice translating between them by focusing on intercept and constant rate of change.

Task A: Match each equation to its y-intercept and slope

Task B: Match tables to equations

Each table shows points on a line. Decide which equation fits, and justify in words.

Choose from these equations:

• y = 2x + 2
• y = -2x - 1

Justification prompts:

• For Table A: What is y when x = 0, and how does y change when x increases by 1?
• For Table B: What is the y-intercept, and is the slope positive or negative?

Task C: Match “graph descriptions” to equations

Instead of a picture, use these descriptions of what you would see on the plane.

• Graph 1: Crosses the y-axis at (0, 3) and goes down 1 for every 2 to the right.
• Graph 2: Crosses the y-axis at (0, -4) and rises 5 for every 1 to the right.
• Graph 3: Crosses the y-axis at (0, 0) and falls 3 for every 1 to the right.

Match to:

• y = (-1/2)x + 3
• y = 5x - 4
• y = -3x

Justification prompt: For each match, state how you used the intercept and the “rise/run” idea from the slope.

Task D: Build an equation and verify with substitution

A line passes through (0, -2) and (4, 6).

• Find the slope m.
• Write the equation in y = mx + b form.
• Verification prompt: Substitute both points into your equation. Do both satisfy it? Write the substitution steps.