Exercises: Solve systems of linear equations - Common Core Math - HS Algebra

A

Warm-Up: Review What You Know

These problems review skills you will need for solving systems of equations.

1.

Solve for $x$ : $3x - 7 = 11$ .

A.

$x = 6$

B.

$x = \frac{4}{3}$

C.

$x = 4$

D.

$x = 18$

2.

Which of the following is the slope-intercept form of the line $2x + y = 5$ ?

A.

$y = 2x + 5$

B.

$y = -2x + 5$

C.

$y = -2x - 5$

D.

$y = \frac{1}{2}x + 5$

3.

Two lines in the coordinate plane have the same slope. What can you conclude?

A.

The lines intersect at exactly one point.

B.

The lines are either parallel (no intersection) or the same line (infinite intersections).

C.

The lines are perpendicular.

D.

The lines intersect at exactly two points.

B

Fluency Practice

Solve each system and verify your solution in both original equations.

$Coordinate plane showing two intersecting lines y = x + 1 and y = -x + 5 meeting at (2, 3)$

1.

The graph shows two intersecting lines. What is the $x$ -coordinate of the solution to the system?

2.

Use substitution to solve the system. What is the value of $y$ ?

$y = 2x - 1$
$3x + y = 9$

3.

Use substitution to solve the system. Express your answer as a fraction. What is the value of $x$ ?

$x - y = 2$
$3x + 2y = 14$

4.

Use elimination to solve the system. What is the value of $x$ ?

$3x + 2y = 11$
$5x - 2y = 13$

5.

Use elimination to solve the system — you must multiply before any variable cancels. What is the value of $y$ ?

$2x + 3y = 7$
$5x - 2y = 8$

C

Varied Practice

These problems use different formats and representations. Apply the most appropriate method.

D

Word Problems

Define variables, write a system of equations, and solve. Interpret your answer in context.

1.

A school store sells pencils for $0.25 each and pens for $0.75 each. On Monday, Marcus bought a total of 8 items and spent $3.00.

How many pencils did Marcus buy? (Write and solve a system of equations.)

$Table comparing Company A (30 per day plus 0.10 per mile) and Company B (20 per day plus 0.20 per mile) car rental rates$

2.

Two car rental companies charge different rates. Company A charges $30 per day plus $0.10 per mile. Company B charges $20 per day plus $0.20 per mile.

1.

At how many miles driven in one day do the two companies charge the same total amount?

2.

If you plan to drive 150 miles in a day, which company is cheaper?

A.

Company A — it has a lower per-mile rate, which matters more at high mileage.

B.

Company B — it has a lower daily rate.

C.

Both companies cost the same at 150 miles.

D.

Company B is always cheaper for more than 100 miles.

3.

A student is deciding whether to solve the system $y = 4x - 3$ and $3x + 2y = 11$ by substitution or by elimination.

Which method would you recommend and why? Solve the system using your chosen method and verify your answer.

E

Error Analysis

Study the student work shown. Identify the error and explain the correct approach.

1.

Taylor solved the system:
$x + y = 7$
$2x - y = 5$

Taylor's work:

Add the equations: $3x = 12$ , so $x = 4$ .
Substitute into the first equation: $4 + y = 7$ , so $y = 3$ .
Check: $4 + 3 = 7$ ✓ (checked only the first equation).
Conclusion: solution is $(4, 3)$ .

Taylor's computation is correct, but the verification is incomplete. What should Taylor do to fully verify the solution?

A.

Substitute $(4, 3)$ into the second equation to confirm $2(4) - 3 = 5$ .

B.

Re-solve using a different method to double-check.

C.

Graph both lines and visually confirm the intersection point.

D.

No further checking is needed — the algebra is error-free.

2.

Jordan solved the system:
$x - 2y = 4 \quad \text{(Equation 1)}$
$3x + y = 7 \quad \text{(Equation 2)}$

Jordan's work:

From Eq. 1: $x = 2y + 4$ .
Substitute into Eq. 2: $3(2y + 4) + y = 7 \Rightarrow 7y = -5 \Rightarrow y = -\frac{5}{7}$ .
Jordan then used the simplified one-variable equation to find $x$ and got a wrong answer.

What mistake did Jordan make in Step 3?

A.

Jordan computed $y = -\frac{5}{7}$ incorrectly.

B.

Jordan should have used elimination instead of substitution.

C.

Jordan substituted back into the derived one-variable equation instead of an original equation.

D.

Jordan should have solved for $y$ first instead of $x$ .

F

Challenge / Extension

These problems require multi-step reasoning. Show all work.

1.

Solve the system using elimination — you must multiply both equations before any variable cancels. What is the value of $x - y$ ?

$2x + 5y = 16$
$3x - 2y = 5$

2.