Linear Equations in Two Variables Pattern - Properties From Equations

Finding slope or intercepts from an equation

This pattern is all about extracting key properties — slope, $y$-intercept, or $x$-intercept — from a linear equation. Sometimes the equation is already in slope-intercept form and the answer is sitting right there. Other times, you need to rearrange, distribute, or combine terms first. The SAT also tests translations (vertical shifts) and equations with fractional coefficients, but the core skill is always the same: get the equation into a useful form and read off what you need.

 

Reading properties directly from $y = mx + b$

When the equation is already in slope-intercept form $y = mx + b$, the slope is $m$ (the coefficient of $x$) and the $y$-intercept is $b$ (the constant). No rearranging needed — just identify the pieces.

The graph of $y = 5x + 12$ crosses the $y$-axis at $(0, b)$. What is the value of $b$?

The equation is already in $y = mx + b$ form. The constant term is $12$, so the $y$-intercept is $12$. That means $b = 12$.

When the equation uses different variable names (like $t$ and $V$, or $n$ and $c$), the same idea applies. Whichever variable plays the role of the output (the one isolated on the left) has its intercept at the constant, and the slope is the coefficient of the input variable.

 

Finding intercepts from standard form $Ax + By = C$

To find the $y$-intercept, set $x = 0$ and solve for $y$. To find the $x$-intercept, set $y = 0$ and solve for $x$. This works for any form of a linear equation.

The equation $7y = 14x - 49$ defines a line. The $y$-intercept is $(0, b)$. What is the value of $b$?

Set $x = 0$: $\;7b = 14(0) - 49 = -49$. Divide by $7$: $b = -7$.

The graph of $-6x - 3y = 18$ has an $x$-intercept at $(x, 0)$. What is the value of $x$?

Set $y = 0$: $\;-6x - 3(0) = 18$, so $-6x = 18$. Divide by $-6$: $x = -3$.

The line $10y - 4x = 20$ has an $x$-intercept at $(c, 0)$ and a $y$-intercept at $(0, d)$. What is the value of $\dfrac{d}{c}$?

$x$-intercept: set $y = 0$. $\;-4x = 20$, so $c = -5$.

$y$-intercept: set $x = 0$. $\;10y = 20$, so $d = 2$.

Therefore $\dfrac{d}{c} = \dfrac{2}{-5} =$ $-\dfrac{2}{5}$.

 

Finding slope by rearranging to $y = mx + b$

When the equation is in standard form $Ax + By = C$, isolate $y$ by moving the $x$-term to the other side and dividing by the coefficient of $y$. The slope is $-\dfrac{A}{B}$.

What is the slope of $9x - 3y = -15$?

Subtract $9x$: $\;-3y = -9x - 15$. Divide by $-3$: $\;y = 3x + 5$. The slope is $3$.

When the equation has $y$ on both sides, collect all $y$-terms first:

What is the slope of $9x + 2y = 5y - 6$?

Subtract $2y$ from both sides: $9x = 3y - 6$. Add $6$: $9x + 6 = 3y$. Divide by $3$: $y = 3x + 2$. The slope is $3$.

When the right side involves distribution, expand and combine like terms before reading the slope:

What is the slope of $y = 6x - \dfrac{1}{5}(10x - 25)$?

Distribute $-\dfrac{1}{5}$: $\;y = 6x - 2x + 5$. Combine: $y = 4x + 5$. The slope is $4$.

 

Intercepts after a vertical translation

When a graph is shifted up by $k$ units, replace $y$ with $(y - k)$ in the original equation. To shift down by $k$, replace $y$ with $(y + k)$. Then find the intercept of the new equation.

The graph of $5x + 8y = 12$ is translated up $3$ units. What is the $y$-coordinate of the $y$-intercept of the resulting graph?

Replace $y$ with $(y - 3)$: $\;5x + 8(y - 3) = 12$. Distribute: $5x + 8y - 24 = 12$. Simplify: $5x + 8y = 36$.

Set $x = 0$: $\;8y = 36$, so $y = \dfrac{36}{8} =$ $\dfrac{9}{2}$.

Alternatively, you can find the original $y$-intercept and just add the shift. From $5x + 8y = 12$ with $x = 0$: $y = \dfrac{12}{8} = \dfrac{3}{2}$. Shift up $3$: $\dfrac{3}{2} + 3 = \dfrac{9}{2}$. Same answer, faster.

 

Intercepts from complex fractional equations

Some hard questions disguise a linear equation behind fractions and parentheses. The strategy is the same: substitute $x = 0$ (for the $y$-intercept) or $y = 0$ (for the $x$-intercept) and solve.

For the equation $3\left(\dfrac{x}{4} - 7\right) = \dfrac{2y}{5}$, what is the $y$-coordinate of its $y$-intercept?

Set $x = 0$: $\;3\left(0 - 7\right) = \dfrac{2y}{5}$, so $-21 = \dfrac{2y}{5}$. Multiply by $5$: $-105 = 2y$. Divide by $2$: $y = -\dfrac{105}{2}$.

What is the $x$-coordinate of the $x$-intercept of $8 - \dfrac{5y}{3} = \dfrac{6x}{7}$?

Set $y = 0$: $\;8 = \dfrac{6x}{7}$. Multiply by $7$: $56 = 6x$. Divide by $6$: $x = \dfrac{56}{6} =$ $\dfrac{28}{3}$.

 

The SLOPE-INTERCEPT Method

1. Identify what's being asked — slope, $y$-intercept, or $x$-intercept.
2. For slope: rearrange to $y = mx + b$ form. Expand any expressions, combine like terms, and isolate $y$. The coefficient of $x$ is the slope.
3. For $y$-intercept: set $x = 0$ and solve for $y$. No need to fully rearrange — just substitute and simplify.
4. For $x$-intercept: set $y = 0$ and solve for $x$.
5. For translations: apply the shift (replace $y$ with $y - k$ for up, $y + k$ for down), then find the requested property from the new equation.

 

Watch Out For

• Sign errors when dividing by a negative. In $-3y = -9x - 15$, dividing by $-3$ flips both signs: $y = 3x + 5$, not $y = -3x - 5$. Double-check every term.
• Confusing slope with its reciprocal. For $Ax + By = C$, the slope is $-\dfrac{A}{B}$, not $-\dfrac{B}{A}$. If you get $\dfrac{1}{3}$ when the answer should be $3$, you likely flipped the fraction.
• Translation direction. Shifting up by $k$ means replacing $y$ with $(y - k)$. The minus sign is counterintuitive — think of it as "the new $y$ must be $k$ less to reach the same old $y$."
• Forgetting to simplify fractional results. After clearing fractions, always reduce. If you get $\dfrac{56}{6}$, simplify to $\dfrac{28}{3}$ before entering your answer.