Solve linear systems strategically

Your feedback matters:

Lesson progressPractice problems 0/5

Difficulty: Intermediate
Estimated time: 37 minutes
Domains: Algebra
Techniques: Method-choiceIntersectionsSubstitutionEliminationRequested-expression

What you’ll learn

Spot when two linear equations are about the same two unknowns.
Read a solution as an ordered pair that makes both equations true.
Choose Desmos, substitution, elimination or a direct combination by looking at how the equations are built.
Give the exact coordinate, ordered pair or expression the question asks for.

Prerequisites

Why this matters on the SAT

Use two facts to find the one value you need

Each two-variable linear equation is one fact about the same two unknowns. Put two of them together and you have a linear system. The SAT might ask for the whole solution, just one coordinate, a mix of both like $x+y$ , or a value from a story, such as how many tickets were sold. The equations won't tell you which method to use. How they're built will.

Solution to the example

Look at the $y$ -terms. Each equation has exactly one $y$ , so subtracting the first equation from the second makes $y$ disappear:

\begin{aligned} (4x+y)-(x+y)&=39-18\\[1.4em] 3x&=21\\[1.4em] x&=7. \end{aligned}

That's A. You never had to find $y$ , because the question only asks for $x$ . The wrong choices are traps worth knowing: B is $y$ , the other coordinate. C is $18$ , the total from the first equation. D is $21$ , the number you have one step before dividing by $3$ .

SAT example

The solution to the system

\begin{aligned} x+y&=18\\[1.4em] 4x+y&=39 \end{aligned}

is $(x,y)$ . What is the value of $x$ ?

A
$7$
B
$11$
C
$18$
D
$21$

Find the shared point, then read the question

Every system here has two linear equations about the same two unknowns, and their lines cross at exactly one point. When both equations look like $ax+by=c$ , you can tell from the coefficients: the lines cross once when one pair $(a,b)$ isn't a constant multiple of the other. In the SAT example, the pairs are $(1,1)$ and $(4,1)$ . No single number turns one into the other, so those lines cross once.

Two lines can also be parallel and never cross, or be the same line. Those cases, with no solution or infinitely many, come next in Model and classify linear systems.

So what does a solution mean? If $(x,y)=(4,-3)$ solves a system, then $x=4$ and $y=-3$ make both equations true. On a graph, $(4,-3)$ is the point where the two lines cross.

That point isn't always your answer, though. Often it's the raw material for one:

What the question asks for	What you enter, using $(4,-3)$
the solution $(x,y)$	$(4,-3)$
just $x$	$4$
just $y$	$-3$
the value of $x+y$	$1$
the value of $2x-y$	$11$

So read the question's last sentence twice: once before you solve and once before you submit. The first read tells you whether you need both coordinates or can stop after one value or one combination. The second read stops you from entering $x$ when it asked for $y$ . Answer the question, not the system.

Check your understanding:

The solution to a system is $(-2,5)$ . What is the value of $3x+y$ , and which coordinate did you use for each variable?

Common mistake:

If you submit the wrong coordinate, you probably read the pair backward or skipped that second read of the question. Write $x=$ and $y=$ next to the point, work out only what’s asked, and make sure your answer has the form the question wants.

Read the solution at a Desmos intersection

Desmos graphs a two-variable linear equation just as it's written, so you don't have to solve it for $y$ first. Take

\begin{aligned} 7x-11y&=-5\\[1.4em] 13x+6y&=70, \end{aligned}

and type 7x-11y=-5 on one line and 13x+6y=70 on the next. Click the intersection, the point where the two lines cross. Desmos shows

(4,3).

So $x=4$ and $y=3$ . If the question asks for $2x+y$ , the point is only your input, and there's one more step:

2x+y=2(4)+3=\boxed{11}.

To be sure you typed and read everything correctly, plug the point back into both original equations:

7(4)-11(3)=-5

and

13(4)+6(3)=70.

Both are true, so the point is right.

Read the graph carefully and keep answers exact

Click the intersection instead of counting grid squares. The two axes can use different scales, so counting can fool you.
Keep the point on screen. If you don't see an intersection, check what you typed first. Then zoom or pan before you decide the lines don't meet.
A graph label can be rounded, so it isn't proof. Seeing $2.667$ doesn't prove the exact value is $\frac83$ . Use fraction conversion when it gives a clear result, compare with exact answer choices, or finish with algebra.
Round only when the question tells you to. When you type your own answer, enter an exact fraction if it fits. If you have to use a decimal, follow the current Bluebook directions for entering answers.

Try it yourself:

Raising the right side of $13x+6y=70$ slides that line up without changing its slope. Change $70$ to $107$ . Before you click, predict: will the crossing point’s $y$ -coordinate go up or down? Then click the new point to check. Hit Reset afterward to get the original system back.

Check your understanding:

Why does the intersection solve both equations, and what extra step do you need if the question asks for $x-y$ ?

Related: For more calculator practice, see Solve systems at intersections. It's optional.

Calculator loads as you approach

Type both equations as written, click where they cross, and match each coordinate to its variable.

Choose the method from the structure

Substitution, elimination, a direct combination and Desmos all do the same job: they trade two equations for something simpler. You only need one of them per problem.

Before you do any algebra, look at the equations and at what the question asks for.

Take the shortcut you can see

If a variable is already alone, as in $x=4y+3$ , or one quick step gets it there, substitute.
If a variable has opposite coefficients, like $2y$ and $-2y$ , or a small multiplier will make them opposites, eliminate.
If adding or scaling the equations builds exactly what the question asks for, use that direct combination. If the question asks for $x+y$ and adding the equations gives $5x+5y=20$ , divide by $5$ and you’re done, with no need to find $x$ or $y$ .

Start with Desmos instead when

The question wants numbers for $x$ and $y$ , and the coefficients are awkward decimals or fractions, like $0.42$ and $0.65$ .
Expanding or rearranging would give you several chances to drop a sign or slip on the arithmetic.
The equations are easy to type in but hard to combine by hand.

When more than one way fits, take the one with the least setup that still keeps your answer exact and makes what’s asked easy to see.

Don’t solve the same system every way. A second method is worth it only as a check.

Check your understanding:

Match each system with the best first method: (1) $y=5x-2$ and $2x+y=19$ ; (2) $3x+4y=10$ and $3x-5y=-8$ ; (3) two equations that are easy to type in but have several unrelated decimal coefficients. Explain each choice without solving.

Substitute when a variable is already alone

Substitution works because equal things can stand in for each other. If one equation says

y=2x+1,

then $y$ and $2x+1$ are the same number, so you can put $2x+1$ in place of every $y$ in the other equation. That leaves a one-variable linear equation. Solve it step by step, then come back for the other variable.

For

\begin{aligned} y&=2x+1\\[1.4em] 3x+y&=16, \end{aligned}

replace the $y$ in the second equation:

\begin{aligned} 3x+(2x+1)&=16\\[1.4em] 5x+1&=16\\[1.4em] 5x&=15\\[1.4em] x&=3. \end{aligned}

Now put $x=3$ into the shorter original equation:

y=2(3)+1=7.

The solution is $(3,7)$ . Check it in the other equation: $3(3)+7=16$ .

Substitution was quickest here because $y$ was already alone. Rearranging both equations or opening a graph would only add setup.

Check your understanding:

For $x=2y-1$ and $x+y=14$ , what should replace $x$ , and what’s the solution?

Common mistake:

Swapping in only part of the expression changes the equation. If $y=2x+1$ , every $y$ becomes the whole $(2x+1)$ . The parentheses matter most after a minus sign: $3x-y$ becomes $3x-(2x+1)=3x-2x-1$ , not $3x-2x+1$ . Solve the one-variable equation, then check the pair in both original equations.

Eliminate when coefficients are ready to cancel

Elimination adds or subtracts whole equations so that one variable drops out. Why is that allowed? Each equation says two amounts are equal. Adding equal amounts to equal amounts gives equal results, so the new equation is still true.

For

\begin{aligned} 3x+2y&=23\\[1.4em] 5x-2y&=17, \end{aligned}

the $y$ -coefficients are already opposites, $2$ and $-2$ . Add the equations:

\begin{aligned} (3x+2y)+(5x-2y)&=23+17\\[1.4em] 8x&=40\\[1.4em] x&=5. \end{aligned}

Put $x=5$ into either original equation:

\begin{aligned} 3(5)+2y&=23\\[1.4em] 2y&=8\\[1.4em] y&=4. \end{aligned}

The solution is $(5,4)$ . Elimination was quickest because you could see the cancellation before rearranging anything.

What if nothing cancels yet? Multiply one equation by the same nonzero number, and do it to every term on both sides. For

\begin{aligned} 2x+3y&=17\\[1.4em] 4x-y&=13, \end{aligned}

multiplying the second equation by $3$ gives $12x-3y=39$ . Now its $-3y$ cancels the $3y$ in the first equation. Adding the two gives $14x=56$ , so $x=4$ , and then $y=3$ .

Check your understanding:

In that second example, why does $4x-y=13$ have to become $12x-3y=39$ and not $12x-3y=13$ ?

Common mistake:

If a variable cancels but the constant never changed, you probably scaled only the variable terms. Write the multiplier outside the whole equation, like $3(4x-y)=3(13)$ , spread it to every term on both sides, and then add or subtract column by column.

Example: Build the expression the question asks for

Worked example

The solution to the system

\begin{aligned} \frac23x+\frac14y&=4\\[1.4em] \frac13x-\frac12y&=-1 \end{aligned}

is $(x,y)$ . What is the value of $6x-\frac32y$ ?

A
$6$
B
$12$
C
$18$
D
$30$

Step 1

Look at what the question wants

The question doesn't ask for $x$ or $y$ on its own. So before you solve anything, see whether adding or scaling the equations can build $6x-\frac32y$ directly.

Step 2

Add the equations

Add the matching terms:

\left(\frac23x+\frac14y\right) + \left(\frac13x-\frac12y\right) =4+(-1).

The $x$ -coefficients add to $1$ , and the $y$ -coefficients add to $-\frac14$ . That gives

x-\frac14y=3.

Step 3

Scale up to the target

Compare $x-\frac14y$ with $6x-\frac32y$ . Six times $x$ is $6x$ , and six times $-\frac14y$ is $-\frac32y$ . So multiply the whole equation by $6$ :

6\left(x-\frac14y\right)=6(3),

which gives

6x-\frac32y=18.

Step 4

Stop once you have it

The expression equals $\boxed{18}$ , so the answer is C.

Want proof the shortcut works? Solve the long way. Clearing the fractions gives $8x+3y=48$ and $2x-3y=-6$ , so $(x,y)=\left(\frac{21}{5},\frac{24}{5}\right)$ . Putting those into $6x-\frac32y$ also gives $18$ . Same answer, but you had to find both fractions to get there.

Common mistake:

Solving for both variables first isn’t wrong, but here it drags you through $\frac{21}{5}$ and $\frac{24}{5}$ , with plenty of room for a slip. When the question asks for an expression, try adding or scaling the original equations before you isolate either variable.

Finish the solution

These decimals would make elimination slow, so this one starts in Desmos. Click the intersection, label the point, and finish the calculation. If you change the calculator, hit Reset to get the original equations back.

Read the point, then finish

Finish the solution

The solution to the system

\begin{aligned} 0.42x+0.65y&=5.31\\[1.4em] 0.78x-0.35y&=5.19 \end{aligned}

is $(x,y)$ . What is the value of $x-y$ ?

First steps

Both original equations are typed in on separate lines.
Their graphs cross at one visible point.

Finish it

Your answer

Calculator loads as you approach

Click the intersection, call the first coordinate x and the second y, then work out x minus y.

Practice problems

For each problem, pick the one method you'd use on test day. Desmos is there if you decide it's the right tool.

Use a variable that’s already alone

Practice problem

The solution to the system

\begin{aligned} y&=3x-4\\[1.4em] x+y&=12 \end{aligned}

is $(x,y)$ . What is the value of $y$ ?

Calculator loads as you approach

Substitution is quickest here. Graph it only if you want to check your answer.

Keep the fraction exact

Practice problem

The solution to the system

\begin{aligned} 3x+2y&=7\\[1.4em] x-2y&=1 \end{aligned}

is $(x,y)$ . What is the value of $x+y$ ?

Your answer

Calculator loads as you approach

Elimination is short here. Graph it if you want to check the crossing point or the final sum.

Read the crossing point in a story

Practice problem

For a print order, let $x$ be the number of black-and-white pages and let $y$ be the number of color pages. The system

\begin{aligned} x+y&=240\\[1.4em] 0.14x+0.52y&=67.80 \end{aligned}

models the page count and printing cost.

How many color pages were produced?

Calculator loads as you approach

Here x is black-and-white pages and y is color pages. Type both equations and click where they cross.

Type it in instead of expanding

Practice problem

The ordered pair $(x,y)$ satisfies

\begin{aligned} 5(3x+2y)-4&=2(x-5y)+2\\[1.4em] 7(x-y)+3&=4(2x+y)+12. \end{aligned}

What is the value of $17(x+y)$ ?

Your answer

Calculator loads as you approach

Typing the equations in saves you from expanding by hand. Click the intersection, then work out the expression.

Finish the lesson

5 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

A solution $(x,y)$ makes both equations true, and it's the point where the two lines cross.
Read what the question asks for before you solve and again before you submit. Answer the question, not the system.
Substitute when a variable is already alone, eliminate when coefficients cancel or scale easily, and use a direct combination when it builds the expression you need.
Use Desmos for the crossing point when typing the equations in saves you awkward or fragile algebra.
Click the intersection instead of reading the grid, match each coordinate to its variable, and keep values exact unless the question says to round.
Check your pair in both original equations, and make sure your final answer means and looks like what was asked.

Related lesson

For deeper practice with graph windows, multiple intersections and your calculator workflow, try Solve systems at intersections.

Next lesson

Model and classify linear systems

Build systems from contexts and decide when two equations have one, no, or infinitely many solutions.

Start next lesson

Practice

Practice this lesson

2,152 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

Start practice