Linear Equations in Two Variables

A linear equation in two variables, such as $y = 2x - 4$ or $3x + 5y = 30$, describes a line in the $xy$-plane. Every point on the line is a solution. The SAT asks you to move between slope-intercept form and standard form, find intercepts, write equations for parallel or perpendicular lines, and set up equations like $8a + 5c = 200$ from a story.

• A point $(p, q)$ is on a line exactly when substituting $x = p$ and $y = q$ makes the equation true.
• In $y = mx + b$, you can read the slope and $y$-intercept directly.
• In $Ax + By = C$, the intercepts are quick: set $x = 0$ to get the $y$-intercept and $y = 0$ to get the $x$-intercept.
• In a model like $8a + 5c = 200$ (adult tickets at $8, child tickets at $5, $200 total), each coefficient is a price per item and the constant is the total. An intercept means "all of one kind": $a = 25$ when $c = 0$.
• Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals.

• Point-slope form: $y - y_1 = m(x - x_1)$.
• Perpendicular slopes multiply to $-1$: slope $\frac{2}{5}$ pairs with $-\frac{5}{2}$.

Find the slope and both intercepts of $6x - 3y = 12$.

1. Slope: $-\frac{A}{B} = -\frac{6}{-3} = 2$.
2. $y$-intercept: set $x = 0$, so $-3y = 12$ and $y = -4$.
3. $x$-intercept: set $y = 0$, so $6x = 12$ and $x = 2$.

Check by rewriting: $-3y = -6x + 12$, so $y = 2x - 4$. Slope $2$, $y$-intercept $-4$. Correct.

Line $k$ is perpendicular to the graph of $3x + 2y = 8$ and passes through $(6, -1)$. What is the $y$-intercept of line $k$?

1. Slope of the given line: $-\frac{3}{2}$.
2. Perpendicular slope: the negative reciprocal, $\frac{2}{3}$.
3. Point-slope form: $y + 1 = \frac{2}{3}(x - 6)$, so $y + 1 = \frac{2}{3}x - 4$.
4. Solve for $y$: $y = \frac{2}{3}x - 5$. The $y$-intercept is $-5$ (the point $(0, -5)$).

Check: at $x = 6$, $y = 4 - 5 = -1$. The line passes through $(6, -1)$, as required.

• Reading the slope of $Ax + By = C$ as $\frac{A}{B}$. The slope is $-\frac{A}{B}$; in $2x + 5y = 20$ it is $-\frac{2}{5}$.
• Half a negative reciprocal. The perpendicular slope to $\frac{2}{5}$ is $-\frac{5}{2}$. Flip the fraction and change the sign.
• Swapping intercepts. In $4x - 3y = 24$, the $y$-intercept is $\frac{24}{-3} = -8$, not $6$ (that is the $x$-intercept).
• Wrong axis in a context. If $n$ is on the horizontal axis, the horizontal intercept is the value of $n$ when the other variable is $0$.
• Forgetting to find a constant first. If a line $3x + ky = 21$ passes through a given point, plug the point in to get $k$ before computing anything else.