Linear Equations and Inequalities

Pentru elevii români: La SAT, ecuațiile liniare apar aproape mereu în context (dolari, mile, ore), iar dreapta se scrie $y = mx + b$, nu $y = mx + n$ ca la noi. Atenție și la separatorul zecimal: în SUA se folosește punctul ($3.5$), iar virgula separă miile ($1{,}200$ înseamnă o mie două sute).

What the SAT tests here

Linear equations and inequalities are the foundation of the Algebra domain of the digital SAT Math section. You will see questions on:

• linear equations in one variable (solve for $x$, or decide how many solutions an equation has);
• linear equations in two variables and linear functions (slope, intercepts, graphs);
• linear inequalities in one or two variables, usually built from a word problem;
• interpreting the numbers in a linear model ("What does 45 represent?").

Some questions are multiple choice; others are student-produced response (SPR) questions where you type the answer yourself.

Key rules and formulas

Slope between $(x_1, y_1)$ and $(x_2, y_2)$:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Forms of a line:

• Slope-intercept: $y = mx + b$ (slope $m$, $y$-intercept $b$)
• Point-slope: $y - y_1 = m(x - x_1)$
• Standard: $Ax + By = C$, with slope $-\frac{A}{B}$ and $y$-intercept $\frac{C}{B}$

Parallel and perpendicular lines: parallel lines have equal slopes; perpendicular lines have slopes whose product is $-1$ (for example, $2$ and $-\frac{1}{2}$).

Number of solutions of $ax + b = cx + d$:

• exactly one solution if $a \neq c$;
• no solution if $a = c$ and $b \neq d$;
• infinitely many solutions if $a = c$ and $b = d$.

Inequalities: solve them like equations, but reverse the inequality sign when you multiply or divide by a negative number. For example, $-3x > 12$ becomes $x < -4$.

In context: in a model such as $C = 45h + 80$, the slope (45) is a rate (dollars per hour) and the constant (80) is a starting value (the fixed fee when $h = 0$).

Worked examples

Example 1. Solve $\frac{2}{3}(x - 6) + 4 = x - 5$.

Distribute: $\frac{2}{3}x - 4 + 4 = x - 5$, so $\frac{2}{3}x = x - 5$. Subtract $x$: $-\frac{1}{3}x = -5$. Multiply by $-3$: $x = 15$. Check: $\frac{2}{3}(9) + 4 = 10$ and $15 - 5 = 10$. ✓

Faster: multiply both sides by 3 first to clear the fraction: $2(x - 6) + 12 = 3x - 15$, which gives $2x = 3x - 15$, so $x = 15$.

Example 2. In the equation $3(2x + k) = 6x - 12$, $k$ is a constant. For what value of $k$ does the equation have infinitely many solutions?

Expand: $6x + 3k = 6x - 12$. The $x$-terms already match, so the equation is true for every $x$ exactly when $3k = -12$, that is, $k = -4$. For any other value of $k$, the equation has no solution.

Example 3. Ana has $60 to spend at a fair. Entry costs $4, and each ride ticket costs $8.50. What is the greatest number of tickets she can buy?

Let $t$ be the number of tickets: $8.50t + 4 \le 60$, so $8.50t \le 56$ and $t \le 6.588\ldots$ Since tickets come in whole numbers, she can buy at most 6 tickets. Rounding to 7 would exceed her budget: $8.50(7) + 4 = 63.50 > 60$.

Test-taking strategy

• Translate words to symbols carefully. "At most" means $\le$; "at least" means $\ge$; "no more than" means $\le$; "fewer than" means $<$.
• Use the built-in Desmos calculator. To solve $\frac{2}{3}(x - 6) + 4 = x - 5$, type y = (2/3)(x-6)+4 and y = x-5 on separate lines and click the intersection point; its $x$-coordinate is the solution.
• Use a slider for constants. For "for what value of $k$..." questions, type the equation with $k$ and let Desmos create a slider. Move it until the two lines overlap (infinitely many solutions) or become parallel (no solution).
• Graph inequalities directly. Typing y < 2x + 3 shades the solution region, which helps when a question asks which point satisfies a system of inequalities.
• SPR answers: you may enter fractions such as 7/2 or decimals such as 3.5. Do not enter mixed numbers, units, or symbols such as $ or %, and follow the on-screen entry directions.
• Plug answer choices back in when algebra feels slow; on a linear equation this check takes seconds.

Common traps

• Forgetting to flip the sign when dividing an inequality by a negative number.
• Rounding the wrong way. In budget or capacity questions, round down; in "how many buses are needed" questions, round up.
• Confusing slope and intercept in context. The number multiplied by the variable is the rate; the constant is the starting amount.
• Answering the wrong quantity. If the question asks for $3x$ or $x + 2$, do not stop at $x$.
• Sign errors in the slope formula. Keep the same order in the numerator and denominator: $(y_2 - y_1)$ over $(x_2 - x_1)$.

Summary

Linear questions reward precise algebra and careful reading. Know the forms of a line, the slope formula, and the three solution cases for $ax + b = cx + d$. Flip inequality signs when you multiply or divide by a negative, round sensibly in context, and interpret slope as a rate and the intercept as a starting value. When in doubt, graph both sides in Desmos and read the intersection.