2.1 Writing Linear Equations in One and Two Variables

2.1 Writing Linear Equations in One and Two Variables

College Board’s Next-Generation ACCUPLACER Advanced Algebra and Functions (AAF) test weights linear equations at 10–15% of the exam — typically 2–3 computer-adaptive items out of 20. Table 11 splits that slice into four skills: creating linear equations in one or two variables, solving them, simplifying linear equations and inequalities, and solving systems of two linear equations. This section is the first skill: write the equation before you solve it.

/practice/accuplacer-advanced-algebraPractice questions with detailed explanations

One-variable equations: ax + b = c

A one-variable linear equation uses the unknown to the first power only. The template is a number times the unknown plus a constant equals another constant. If a taxi charges a $4.50 flag drop plus $2.00 per mile and the fare is $16.50, you are not yet isolating m. You are creating:

4.50 + 2.00m = 16.50

That is ax + b = c with a = 2, x = m, b = 4.50, and c = 16.50. AAF stems hide those three numbers in ordinary language:

• A gym charges $35 plus $8 per class, and a member’s bill is $83: 35 + 8c = 83.
• A phone plan is $20 plus $0.15 per text, totaling $32: 20 + 0.15t = 32.
• A plumber charges $75 to arrive plus $45 per hour, and the invoice is $210: 75 + 45h = 210.

Translation checklist

1. Name the unknown (miles, classes, hours).
2. Identify the rate — the coefficient of the unknown.
3. Identify the fixed amount — the constant term.
4. Identify the total — the other side of the equation.
5. Do not park the rate on the right-hand side: $8 per class is 8c, not the 83.

If two unknown quantities appear (s small bottles and ℓ large bottles), you have left one-variable writing and entered two-variable writing, possibly a system (Section 2.4).

Two-variable equations: three forms you must convert

When two quantities vary together, AAF expects fluency in three equivalent forms.

Slope is rise over run: m = (y₂ − y₁) / (x₂ − x₁) provided x₂ ≠ x₁. A horizontal line has m = 0 and equation y = k. A vertical line has undefined slope and equation x = k. Vertical lines fail the vertical-line test, so they are not functions and they cannot be written as y = mx + b. Treating x = 4 as a function of x is a standard AAF trap.

Worked example: table (2, 5) and (4, 11) → y = 3x − 1

A table lists two points on a line: (2, 5) and (4, 11).

1. Compute slope: m = (11 − 5) / (4 − 2) = 6 / 2 = 3.
2. Write point-slope from (2, 5): y − 5 = 3(x − 2).
3. Expand: y − 5 = 3x − 6, so y = 3x − 1.
4. Check the unused point: 3(4) − 1 = 12 − 1 = 11. Both ordered pairs satisfy the equation.

The y-intercept is (0, −1). The x-intercept solves 0 = 3x − 1, so x = 1/3 and the point is (1/3, 0). Swapping those intercepts is the most common writing error on this skill: the constant sitting alone in y = mx + b is the y-intercept, not the x-intercept.

Starting from (4, 11) instead, y − 11 = 3(x − 4) simplifies to the same y = 3x − 1. If two different pairs of points produce two different slopes, the data are not collinear and no single linear equation fits both. When a table offers three points, compute m twice before you commit.

Standard form of the same line: 3x − y = 1 (or −3x + y = −1). Prefer integers with A positive and gcd(|A|, |B|, |C|) = 1 unless the item specifies otherwise.

Worked example: from a graph and from intercepts

A graph crosses the y-axis at (0, 4) and also passes through (6, 1).

• The intercept supplies b = 4, so the skeleton is y = mx + 4.
• m = (1 − 4) / (6 − 0) = (−3) / 6 = −1/2.
• Slope-intercept: y = (−1/2)x + 4.

Clear the fraction for standard form. Multiply through by 2: 2y = −x + 8, then x + 2y = 8. Here A = 1, B = 2, C = 8.

If the same graph is a vertical line through (−3, 2) and (−3, 7), the equation is x = −3. There is no finite m. If the graph is horizontal through (−3, 2) and (5, 2), then m = 0 and y = 2.

Worked example: converting among the three forms

Start with point-slope y − 4 = 2(x + 3). The encoded point is (−3, 4) and the slope is 2.

• Slope-intercept: y − 4 = 2x + 6 → y = 2x + 10. Slope 2, y-intercept (0, 10).
• Standard form: −2x + y = 10. Multiply by −1 if the item wants A > 0: 2x − y = −10. Both are equivalent; follow the stem.

Going the other way, start with 3x − 6y = 12 and solve for y:

−6y = −3x + 12 → y = (1/2)x − 2.

Slope is 1/2, y-intercept is −2. The coefficient of x in standard form is A = 3, which is not the slope. Always isolate y before you read m and b.

Word contexts in two variables

A vendor sells small bottles for $2 and large bottles for $5. Revenue of $80 is already standard form: 2s + 5ℓ = 80. If the vendor always sells 3 large bottles for every 1 small bottle, you add a second equation and move into systems (Section 2.4). AAF may stop at “write the equation,” or it may treat one quantity as independent and ask for a function, such as R(n) = 2n for revenue from n small bottles when large bottles are held fixed.

A temperature model F = (9/5)C + 32 is slope-intercept: slope 9/5 (Fahrenheit degrees per Celsius degree) and F-intercept 32 when C = 0. Standard form: multiply by 5 to get 5F = 9C + 160, then 9C − 5F = −160. You do not need to memorize the conversion; you need to recognize slope as the multiplier of the independent variable and the constant as the output when the input is 0.

Traps that cost CAT items

• Mixing intercepts. In y = 3x − 1, −1 is the y-coordinate of the y-intercept. The x-intercept is 1/3, not −1.
• Using run over rise. For (2, 5) and (4, 11), m = 6/2 = 3, not 2/6.
• Dropping the distributed term in point-slope. 3(x − 2) is 3x − 6, not 3x − 2.
• Forcing a vertical line into y = mx + b. x = 4 is a line, not a function, and its slope is undefined.
• Scaling only one side. y = (1/2)x + 3 becomes 2y = x + 6, not 2y = x + 3.

When three table points are given — say (0, 2), (3, 8), and (6, 14) — the pairwise slopes are all 2, so y = 2x + 2 is justified. If the third point broke the pattern, the relationship would not be linear and you would not invent a single y = mx + b.

Writing is skill one of four on Table 11. Next you isolate the variable, then you handle inequalities, then you solve systems. Keep the graphing toolkit one click away at slope-intercepts-graphs.