SAT Linear equations · two variables

Slopes, intercepts, and interpreting coefficients of a line.

7% of MathMath · Algebra5 question types

~3per test

How to score it

Perpendicular = negative reciprocal slope; parallel = same slope.
Rewrite Ax + By = C into y = mx + b before reading off the slope.
For interpretation, attach units to the coefficient and read it as a rate.

Common traps

Forgets to flip the sign or reciprocal on “perpendicular.”
Interprets the wrong coefficient (the one that's a product).
Reads slope straight from standard form without rearranging.

The 5 question types, with real examples

Slope: parallel / perpendicular

“Line k is defined by [equation]. Line j is perpendicular to k. What is the slope of j?”

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From the question bankMedium

Line j is defined by $5y - 15x = 25$ . Line k is parallel to line j in the xy-plane. What is the slope of line k?

A $-3$
B $\displaystyle \frac{1}{3}$
C $3$ ✓
D $5$

Why C

Parallel lines have equal slopes. Rewriting $5y - 15x = 25$ in slope-intercept form: $5y = 15x + 25$ , so $y = 3x + 5$ . The slope of line j is 3, so the slope of line k is also 3.

Interpret a coefficient in context

“Which statement is the best interpretation of [coefficient] in this context?”

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From the question bankMedium

A bakery charges a setup fee plus an additional cost per dozen cookies ordered. The total cost P, in dollars, for ordering d dozen cookies is given by $P = 20 + 6d$ . Which statement is the best interpretation of 6 in this context?

AThe total cost increases by $26 for each additional dozen cookies ordered.
BThe bakery charges $5 per dozen cookies ordered.
CThe bakery charges $6 as a setup fee.
DThe bakery charges $6 per dozen cookies ordered.✓

Why D

In the equation $P = 20 + 6d$ , the coefficient 6 is multiplied by d (the number of dozens of cookies). This means for each dozen cookies, $6 is added to the total cost.

Graph → identify the equation

“Which equation defines line ℓ shown in the graph?”

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From the question bankMedium

In the xy-plane, line k is defined by $6x + 2y = 18$ . What is the slope of line k?

A $-3$ ✓
B $-2$
C $2$
D $3$

Why A

Rewriting the equation in slope-intercept form: $2y = -6x + 18$ , so $y = -3x + 9$ . The slope is -3.

Find missing parameter from a graphed line / table

“[table or graph through (a,b) and (c,d), equation given parametrically] -> solve for the parameter.”

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From the question bankMedium

In the xy-plane, the graph of the equation $3x - 6y = 18$ has slope m. What is the value of m?

A $-2$
B $\displaystyle \frac{1}{2}$ ✓
C $2$
D $3$

Why B

Rewriting $3x - 6y = 18$ in slope-intercept form: $-6y = -3x + 18$ , so $\displaystyle y = \frac{1}{2} x - 3$ . The slope is 1/2.

Equation-of-line from two points (write the equation)

“Which [equation/table] represents this situation / the given equation?”

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From the question bankMedium

In the xy-plane, line t passes through the origin and has a slope of $\displaystyle \frac{3}{4}$ .. Which equation defines line t?

A $3x - 4y = 0$ ✓
B $3x + 4y = 0$
C $4x - 3y = 0$
D $4x + 3y = 0$

Why A

A line through the origin with slope 3/4 has equation $\displaystyle y = \frac{3}{4}$ x. Multiplying both sides by 4 gives $4y = 3x$ , which rearranges to $3x - 4y = 0$ .

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SAT Linear equations · two variables

How to score it

Common traps

The 5 question types, with real examples

Slope: parallel / perpendicular

Interpret a coefficient in context

Graph → identify the equation

Find missing parameter from a graphed line / table

Equation-of-line from two points (write the equation)

More Algebra skills

Knowing the pattern is step one. Drilling it is how you score.