SAT Systems of two linear equations

Solve two equations at once — often for a clever combination.

6% of MathMath · Algebra4 question types

~3per test

How to score it

If they ask for x + y or 2x − y, try adding or subtracting the equations directly — you may skip solving entirely.
Context problems: define two variables, write two equations, eliminate.
“No solution” = same slope, different intercept (parallel lines).

Common traps

Solves fully when adding the equations gives the answer instantly.
“No solution” treated as “solve it” instead of matching coefficients.
Mixes up which variable the question wants.

The 4 question types, with real examples

Solve, return a derived value

“The solution is (x, y). What is the value of [x + y / xy / 2x − y]?”

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From the question bankMedium

$y = 5x - 7$ $4x + y = 29$ The solution to the given system of equations is (x, y). What is the value of 3x?

A $4$
B $8$
C $-12$
D $12$ ✓

Why D

Substituting $y = 5x - 7$ into the second equation yields $4x + 5x - 7 = 29$ , or $9x = 36$ . Solving gives $x = 4$ . Therefore, $3x = 12$ .

Context: two prices, two purchases

“[Person A bought …; Person B bought …] How many [items] … ?”

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From the question bankMedium

Sophia bought 3 notebooks and 5 pens for $18.45. Diego bought 2 notebooks and 3 pens for $11.70. How many notebooks are in Diego's purchase?

A $3$
B $5$
C $2$ ✓
D $1$

Why C

Let n be the price of one notebook and p be the price of one pen. From the given information: $3n + 5p = 18.45$ and $2n + 3p = 11.70$ . Solving this system yields $n = 2.85$ and $p = 1.80$ . The question asks how many notebooks are in Diego's purchase, which is stated directly in the problem as 2.

Constant for no / infinite solutions

“If the system has no solution, what is the value of [parameter]?”

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From the question bankHard

$4x + 3y = 24$ $8x - ky = 12$ For what value of k will the system of equations above have no solution?

A $-12$
B $-6$ ✓
C $6$
D $3$

Why B

For no solution, the lines must be parallel but distinct. The x-coefficients have ratio 8/4 = 2, so multiplying the first equation by 2 gives $8x + 6y = 48$ . For parallel lines, we need 8x - ky = 12 to have $-k = 6$ , so $k = -6$ . The constant terms differ $(48 \ne 12)$ , confirming no solution.

Identify a point that lies on both lines

“For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?”

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From the question bankMedium

$y = 2x - r$ $y = x + 7$

For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?

A $(7 + r, 14 + r)$ ✓
B $(7 - r, 14 - r)$
C $(7 + r, 7)$
D $\displaystyle (\frac{7 + r}{2}, \frac{14 + r}{2})$

Why A

Setting the equations equal: $2x - r = x + 7$ , so $x = 7 + r$ . Substituting into $y = x + 7$ gives $y = (7 + r) + 7 = 14 + r$ .

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SAT Systems of two linear equations

How to score it

Common traps

The 4 question types, with real examples

Solve, return a derived value

Context: two prices, two purchases

Constant for no / infinite solutions

Identify a point that lies on both lines

More Algebra skills

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