Linear Equations in Two Variables Pattern - Properties From Data

Finding slope, intercepts, or equations from a graph or table

This pattern gives you a line — either drawn on a coordinate plane or described by a table of values — and asks you to extract a property: the slope, an intercept, or the full equation. The core skills are reading coordinates off a graph, computing slope from two points, and recognizing the $y = mx + b$ form.

 

Reading Intercepts from a Graph

The simplest version: a line is drawn and you're asked for the $x$-intercept or $y$-intercept. The $y$-intercept is where the line crosses the $y$-axis (the point where $x = 0$), and the $x$-intercept is where it crosses the $x$-axis (where $y = 0$). Just read the coordinates off the graph.

What is the y-intercept of the graphed function?

The line crosses the $y$-axis at the point where $x = 0$. Reading the graph, that crossing happens at $y = -2$. The $y$-intercept is $\boldsymbol{(0, -2)}$.

The $x$-intercept works the same way — find where the line crosses the $x$-axis:

What is the x-intercept of the line?

The line crosses the $x$-axis at the point where $y = 0$. From the graph, that's at $x = -4$. The $x$-intercept is $\boldsymbol{(-4, 0)}$.

A common trap: the SAT offers the $y$-intercept as a wrong answer when asking for the $x$-intercept, and vice versa. Always check which axis the question is asking about.

 

Finding the Equation from a Graph

When the question asks for the equation of a graphed line, you need two things: the slope and the $y$-intercept. Read the $y$-intercept directly, then pick two clear grid points to calculate slope using $m = \dfrac{y_2 - y_1}{x_2 - x_1}$.

Which equation represents the line shown in the xy-plane?

The line crosses the $y$-axis at $(0, -2)$, so $b = -2$. It also passes through $(-3, 0)$. Calculate the slope:

$$m = \frac{-2 - 0}{0 - (-3)} = \frac{-2}{3}$$

The equation is $\boldsymbol{y = -\dfrac{2}{3}x - 2}$.

Here's another — this line has a steeper negative slope:

Which of the following is an equation of the line?

The $y$-intercept is $(0, 6)$, so $b = 6$. The line also passes through $(2, 0)$. The slope is:

$$m = \frac{0 - 6}{2 - 0} = \frac{-6}{2} = -3$$

The equation is $\boldsymbol{y = -3x + 6}$.

 

Finding the Equation from a Table

When the data comes in a table instead of a graph, the approach is the same: pick two rows, compute the slope, then use one point to find the $y$-intercept. If the table includes the point where $x = 0$ (or whatever the input variable is), you can read the intercept directly.

$t$ (minutes)	$T$ (°C)
$0$	$90$
$1$	$86$
$2$	$82$

An object is cooling in a room. The table shows the object's temperature $T$ at time $t$ minutes, with a linear relationship. Which equation describes this relationship?

The table gives us the point $(0, 90)$, so the $T$-intercept is $90$. Use any two points for the slope:

$$m = \frac{86 - 90}{1 - 0} = \frac{-4}{1} = -4$$

The equation is $\boldsymbol{T = -4t + 90}$.

 

Slope with Unknown Constants

The SAT sometimes puts an unknown constant into the table values. The trick: the constant cancels when you compute the slope, because slope only depends on differences between values.

$x$	$v$
$-4$	$c + 100$
$0$	$c + 80$
$4$	$c + 60$

The relationship between $x$ and $v$ is linear, where $c$ is a constant. What is the slope of the line?

Pick the first two rows. The slope is:

$$m = \frac{(c + 80) - (c + 100)}{0 - (-4)} = \frac{-20}{4} = -5$$

The $c$'s cancel out. The slope is $\boldsymbol{-5}$.

Here's a similar idea with a different setup — the constant appears in the $x$-values instead:

$h$ (hours)	$d$ (miles)
$1$	$300 - 2b$
$3$	$180 - 2b$
$5$	$60 - 2b$

A car travels at constant speed. The table shows distance $d$ from home after $h$ hours, where $b$ is a constant. What is the slope?

Use the first two rows:

$$m = \frac{(180 - 2b) - (300 - 2b)}{3 - 1} = \frac{-120}{2} = -60$$

Again, the $-2b$ terms cancel. The slope is $\boldsymbol{-60}$.

 

Harder Variations

On hard questions, the SAT may give you a graph but ask about a different function defined in terms of the graphed one. You need to find the equation of the graphed line first, then do algebra.

The graph shows $y = g(x) - 8$. Which equation defines $g$?

First, find the equation of the graphed line. It passes through $(0, -1)$ and $(2, 5)$:

$$m = \frac{5 - (-1)}{2 - 0} = \frac{6}{2} = 3$$

So the graphed line is $y = 3x - 1$. Since the graph represents $y = g(x) - 8$:

$$g(x) - 8 = 3x - 1$$ $$g(x) = 3x - 1 + 8 = 3x + 7$$

Therefore $\boldsymbol{g(x) = 3x + 7}$.

Another hard variant uses a table with constants and asks you to find a new point on the line:

$x$	$y$
$c$	$-8$
$c - 6$	$16$

A line passes through the points in the table. If $(c + 2, d)$ is also on the line, what is $d$?

Find the slope: $m = \dfrac{16 - (-8)}{(c - 6) - c} = \dfrac{24}{-6} = -4$.

The point $(c + 2, d)$ is $2$ units to the right of $(c, -8)$. With slope $-4$, each unit right means $4$ units down:

$$d = -8 + (-4)(2) = -8 - 8 = -16$$

Therefore $\boldsymbol{d = -16}$.

The SAT also tests this without any visual — just a slope and one point:

Line $L$ has slope $\dfrac{3}{4}$ and $x$-intercept $(8, 0)$. What is the $y$-coordinate of the $y$-intercept?

Use slope-intercept form $y = mx + b$ with $m = \dfrac{3}{4}$. Plug in the point $(8, 0)$:

$$0 = \frac{3}{4}(8) + b = 6 + b$$

So $b = -6$. The $y$-coordinate of the $y$-intercept is $\boldsymbol{-6}$.

 

The READ-COMPUTE-BUILD Method

1. Read — identify two clear points from the graph or table. On a graph, use points where the line crosses grid intersections. In a table, pick any two rows.
2. Compute — calculate the slope: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. If the question only asks for slope, you're done.
3. Build — if the question wants the full equation, use $y = mx + b$. If one of your points has $x = 0$, you already know $b$. Otherwise, plug any point into $y = mx + b$ and solve for $b$.

 

Watch Out For

• Mixing up $x$- and $y$-intercepts. The SAT deliberately offers the $y$-intercept when asking for the $x$-intercept. The $y$-intercept has the form $(0, b)$; the $x$-intercept has the form $(a, 0)$. Check which one the question asks for.
• Slope sign errors. When both coordinates are negative or you're subtracting a negative, it's easy to drop a sign. Write out the subtraction fully: $\frac{-2 - 0}{0 - (-3)}$, not $\frac{2}{3}$.
• Constants that look scary but cancel. When a table has expressions like $c + 100$ and $c + 80$, the slope formula only uses differences, so the constant disappears. Don't try to solve for the constant — just compute the differences.
• Confusing the graphed function with the asked function. If the question says the graph shows $y = f(x) - k$ and asks for $f(x)$, you must add $k$ back. Read the question stem carefully to see exactly what's being graphed versus what's being asked for.