15.3 Translating Between Graphs and Line Equations

15.3 Translating Between Graphs and Line Equations

Quick Answer: Read a line’s y-intercept from the crossing of the y-axis and its slope from rise/run between two points on the graph; then write y = mx + b. Horizontal lines are y = k; vertical lines are x = h. Convert standard form Ax + By = C into slope-intercept by solving for y, and reverse the process by rearranging and clearing fractions. PERT coordinate items often ask you to pick which equation matches a described graph—or which graph matches an equation—without needing a full drawing if you lock m and b.

This section ties together slope, intercepts, and form conversions so you can move freely among a verbal graph description, a sketch, and algebraic forms. The algebra is the same as in 15.1; the skill is translation—matching representations under time pressure.

Reading Slope and Intercept from a Graph Description

When a problem describes a graph (or you sketch one):

1. Find the y-intercept point (0, b).
2. Find a second clear point (x, y) with integer coordinates if possible.
3. Compute m = (y − b)/x, or use any two points with the slope formula.
4. Write y = mx + b.

Worked Example 1 — Graph description → equation

A line crosses the y-axis at (0, −1) and passes through (4, 5). Which equation matches?

m = (5 − (−1))/(4 − 0) = 6/4 = 3/2.

Equation: y = (3/2)x − 1.

Check (4, 5): (3/2)(4) − 1 = 6 − 1 = 5. ✓

Worked Example 2 — Count rise/run on a grid

Suppose the line goes through (0, 2) and (3, 0) on a grid.

m = (0 − 2)/(3 − 0) = −2/3, b = 2 → y = −(2/3)x + 2.

Notice the x-intercept is 3: set y = 0 → 0 = −(2/3)x + 2 → (2/3)x = 2 → x = 3. Matching both intercepts is a fast way to eliminate wrong choices.

Worked Example 3 — Intercepts only

A line has x-intercept 6 and y-intercept −3. Find its equation.

Points: (6, 0) and (0, −3).

m = (0 − (−3))/(6 − 0) = 3/6 = 1/2.

Using b = −3: y = (1/2)x − 3.

Standard form option: multiply by 2 → 2y = x − 6 → x − 2y = 6.

Matching a Graph to an Equation (and vice versa)

Use a two-check method:

1. Intercept check: Does the equation’s b match the graph’s y-intercept? Do the x-intercepts match when computed?
2. Slope check: Is the line rising or falling as expected? Is the steepness about right (steep |m| large)?
3. Extra point: Plug a non-intercept lattice point from the graph into the equation.

Worked Example 4 — Eliminate options with b

Graph crosses y-axis at −4 and falls gently left-to-right. Options:

A) y = 2x − 4
B) y = −(1/4)x − 4
C) y = −(1/4)x + 4
D) y = 4x − 4

Same intercept −4 narrows to A, B, D. “Falls” means negative slope → only B. Steep positive 2 or 4 contradicts “gently falls.”

Worked Example 5 — Verify with a second point

Claimed equation y = 2x + 1; graph appears to pass through (3, 7).

2(3) + 1 = 7. ✓ Also b = 1 matches a y-intercept of 1. If the graph showed (3, 6), the equation would be wrong (perhaps y = 2x or y = x + 1).

Horizontal and Vertical Lines

Worked Example 6 — Horizontal through a point

Line through (5, −3) horizontal → y = −3. Every point has y-coordinate −3. Slope between (5, −3) and (0, −3) is 0.

Worked Example 7 — Vertical through a point

Line through (−2, 7) vertical → x = −2. Distance formulas still work; slope-intercept form does not apply. Perpendicular to this line would be any horizontal line y = constant.

Worked Example 8 — Identify from description

“The graph is a vertical line three units to the right of the y-axis.” → x = 3.

“The graph is horizontal and four units below the x-axis.” → y = −4.

Converting Between Forms

Standard → slope-intercept

Ax + By = C with B ≠ 0:

By = −Ax + C → y = (−A/B)x + C/B

So m = −A/B and b = C/B.

Worked Example 9 — Standard to slope-intercept

Convert 3x − 6y = 12 to slope-intercept form.

−6y = −3x + 12 → divide by −6: y = (3/6)x − 12/6 → y = (1/2)x − 2.

Alternatively divide the original equation by 3 first: x − 2y = 4 → −2y = −x + 4 → y = (1/2)x − 2. Same result.

Worked Example 10 — Fractions and signs

Convert 2x + 5y = −10.

5y = −2x − 10 → y = −(2/5)x − 2.

m = −2/5, b = −2. Trap: writing +2 for b by dropping the sign on −10/5.

Slope-intercept → standard (integer coefficients)

Start with y = mx + b. Move x terms left: −mx + y = b, then multiply through to clear denominators and optionally make A positive.

Worked Example 11 — Clear fraction

Convert y = −(3/4)x + 5 to standard form with integer coefficients.

y + (3/4)x = 5
Multiply by 4: 4y + 3x = 20 → 3x + 4y = 20.

Check: solve back → 4y = −3x + 20 → y = −(3/4)x + 5. ✓

Worked Example 12 — Negative slope already integer

y = −2x + 7 → 2x + y = 7 (add 2x to both sides). Some texts prefer Ax + By + C = 0 form: 2x + y − 7 = 0; PERT usually accepts Ax + By = C.

Putting Translation Skills Together

Worked Example 13 — Full McCann-style match

A line passes through the origin and the point (2, −6). Which is correct?

m = (−6 − 0)/(2 − 0) = −3, b = 0 → y = −3x.

Equivalent standard form: 3x + y = 0. Horizontal/vertical options are wrong because the line clearly tilts. Any equation with nonzero intercept fails the “through origin” test: plug (0, 0) must satisfy the equation.

Worked Example 14 — Parallel graph, different intercept

Graph A: y = 2x + 1. Graph B is parallel and crosses the y-axis at −4.

Same slope 2, b = −4 → y = 2x − 4. In standard form: −2x + y = −4, or 2x − y = 4.

Worked Example 15 — Which form is fastest?

Test-Day Checklist

1. Underline whether the question wants slope-intercept, standard, or “which graph.”
2. For graph → equation: lock b, then m, then write y = mx + b; convert only if options are standard form.
3. For equation → graph: plot b, use m as rise/run to a second point, glance at x-intercept as a check.
4. Instantly classify horizontal (y = k) and vertical (x = h) before forcing slope-intercept.
5. When converting, solve for y or clear denominators in one disciplined pass—mixed steps create sign distractors.

If you can translate among picture, description, y = mx + b, and Ax + By = C without hesitation, you have finished the coordinate-plane line skills on the McCann PERT Mathematics outline. Pair this fluency with distance and midpoint from 15.2 and you can handle mixed geometry-algebra items as structured substitutions rather than guesswork.