How do you solve a system of equations on the ACT?

Use substitution when one variable is already isolated, or elimination when both equations are in ax + by = c form. Solve for one variable, back-substitute to find the other, and report the ordered pair (x, y) that satisfies both equations.

When does a system of equations have no solution?

A system has no solution when the two lines are parallel: they have the same slope but different y-intercepts, so the lines never intersect and no point satisfies both equations.

What does infinitely many solutions mean?

Infinitely many solutions means the two equations describe the exact same line, with identical slope and identical y-intercept. One equation is a constant multiple of the other, so every point on the line works.

Should I use substitution or elimination?

Use substitution when a variable is already isolated, like y = 2x + 1. Use elimination when both equations are in standard form and adding or subtracting them cancels a variable directly.

CramClub • Systems of Equations — ACT® Guide & Practice

Isolate one variable, then replace it in the other equation.

Add or subtract the equations to cancel one variable.

Same slope, different intercept

Same slope AND same intercept

What a System Asks

A system of equations is two (or more) equations that share the same variables, and a solution is a set of values that makes every equation true at the same time. For two linear equations in $x$ and $y$ , each equation is a straight line, and the solution is the single point where the lines intersect. On the ACT you will solve systems directly, count how many solutions a system has, and — most often — build a system from a word problem.

Method 1 — Substitution

Substitution is fastest when one variable is already isolated or is easy to isolate.

Solve $y = 2x + 1$ and $3x + y = 16$ .

3x + (2x + 1) &= 16 \\ 5x + 1 &= 16 \\ 5x &= 15 \\ x &= 3 \end{aligned}$$ Back-substitute: $y = 2(3) + 1 = 7$. The solution is $(3, 7)$. Always confirm in the other equation: $3(3) + 7 = 16$. ✓ ## Method 2 — Elimination Elimination shines when both equations are in the form $ax + by = c$. Add or subtract the equations so one variable cancels; scale one equation first if needed. **Solve** $3x + 2y = 12$ and $5x - 2y = 4$. The $y$-terms are already opposites, so add the equations: $$\begin{aligned} (3x + 2y) + (5x - 2y) &= 12 + 4 \\ 8x &= 16 \\ x &= 2 \end{aligned}$$ Then $3(2) + 2y = 12$, so $2y = 6$ and $y = 3$. The solution is $(2, 3)$. If the coefficients are not opposites, multiply one equation by a constant first. To solve $4x + 3y = 10$ with $x - y = 6$, multiply the second by 3 to get $3x - 3y = 18$; adding eliminates $y$. ## How Many Solutions? A favorite ACT twist gives a system with an unknown constant and asks how many solutions it has. Put both equations in slope-intercept form and compare. | Lines | Slopes and intercepts | Solutions | |---|---|---| | Cross once | Different slopes | **Exactly one** | | Same line | Same slope **and** same intercept | **Infinitely many** | | Parallel | Same slope, **different** intercept | **None** | **Example.** For what value of $k$ does $y = kx - 2$ and $6x - 3y = 9$ have no solution? Rewrite the second as $y = 2x - 3$. Parallel lines share a slope, so $k = 2$; the intercepts already differ ($-2 \ne -3$), so there is genuinely no solution. ## Building a System from a Word Problem The highest-value skill is translation. Most ACT system word problems give you two facts about two unknowns — usually a **count** and a **total**. > Adult tickets cost \$12 and student tickets cost \$8. A group buys 20 tickets for \$190. How many adult tickets did they buy? Let $a$ be adult tickets and $s$ be student tickets. The count gives $a + s = 20$; the money gives $12a + 8s = 190$. Solve the first for $s = 20 - a$ and substitute: $$\begin{aligned} 12a + 8(20 - a) &= 190 \\ 12a + 160 - 8a &= 190 \\ 4a &= 30 \\ a &= 7.5 \end{aligned}$$ On a real ACT the numbers usually land on whole tickets; a non-integer is a cue to recheck the setup or your arithmetic. (With a \$186 total instead, $4a = 26$… still not whole — the takeaway is to *always verify* the total, because the ACT rewards careful setup.) ## A Time-Saving Shortcut Sometimes the ACT wants a **combination** like $x + y$, not the two variables separately. If $5x + 2y = 23$ and $2x + 5y = 19$, adding gives $7x + 7y = 42$, so $x + y = 6$ in a single step — no need to isolate anything. ## Choosing Your Method - **A variable is already isolated** → **substitution**. - **Both equations are in $ax + by = c$ form** → **elimination**. - **"How many solutions?"** → compare slopes and intercepts; do not solve. - **The question wants $x + y$ or $x - y$** → try adding or subtracting the equations directly. ## Common Mistakes to Avoid - Solving for one variable and stopping — a system's answer is an ordered pair. - Scaling only one side of an equation when preparing to eliminate. - Confusing "no solution" (parallel) with "infinitely many" (same line). - Mislabeling variables in a word problem, so the count and total equations get swapped.

Solve the system $y = x - 4$ and $2x + y = 11$ .

Solve the system $2x + 3y = 13$ and $4x - 3y = 5$ .

Solve the system $3x + 4y = 18$ and $2x + y = 7$ .

A concession stand sells hot dogs for $4 and pretzels for $3. In one hour it sells 25 items for a total of $86. How many hot dogs did it sell?

For what value of $k$ does the system $y = kx + 1$ and $8x - 2y = 6$ have no solution?

If $(x, y)$ satisfies $5x + 3y = 28$ and $3x + 5y = 20$ , what is the value of $x - y$ ?

Solving for one variable and stopping. Finding $x = 3$ and bubbling it in leaves half the solution.

A system in two variables has an ordered-pair solution $(x, y)$ . Back-substitute to get the second coordinate unless the question asks for only one variable.

Scaling only one side of an equation, writing $8x + 4y = 28$ from $2x + y = 7$ but forgetting to multiply the right side.

Multiply every term on both sides by the same factor: $4(2x + y) = 4(7)$ gives $8x + 4y = 28$ .

Confusing "no solution" with "infinitely many." Both have equal slopes, so they are easy to mix up.

Check the intercepts too: equal slope AND equal intercept → infinitely many; equal slope, different intercept → no solution.

Swapping the count and total equations in a word problem, e.g. writing $4h + 3p = 25$ .

Keep units straight: the count equation adds quantities ( $h + p = \text{items}$ ); the money equation multiplies price by quantity ( $4h + 3p = \text{dollars}$ ).

What is the value of $x$ in the solution to the system $y = x + 3$ and $2x + y = 12$ ?

If $x + y = 15$ and $x - y = 3$ , what is the value of $y$ ?

A movie theater sells adult tickets for $11 and child tickets for $7. One showing sells 30 tickets for a total of $274. How many adult tickets were sold?

For what value of $c$ does the system $y = 3x - 5$ and $6x - 2y = c$ have infinitely many solutions?

The system $2x + 5y = 9$ and $6x + ky = 27$ has infinitely many solutions. What is the value of $k$ ?

If $6x + 5y = 41$ and $5x + 6y = 36$ , what is the value of $x + y$ ?