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Systems of Linear Equations

A free Algebra I lesson from the “Systems of Linear Equations and Inequalities” unit, with a worked example and practice problems including step-by-step solutions.

A system of equations asks for values that make both equations true at the same time. On a graph, the solution is the point where the lines intersect. In Systems of Linear Equations and Inequalities, students need more than a memorized rule: they need to recognize the structure, select a method, carry out the algebra cleanly, and interpret the answer in a graph, table, equation, or real context. The expanded practice now mixes skill fluency, transfer questions, and cumulative review so the lesson builds durable Algebra I readiness.

What you'll learn

Why it matters: Systems model moments when two plans meet, such as equal costs, crossing paths, matching supply and demand, or two constraints on one project.

Worked example

Problem. Solve x + y = 10 and y = 4.

  1. Substitute y = 4 into x + y = 10.
  2. x + 4 = 10, so x = 6.
  3. The solution is (6, 4).
  4. Connect the result back to Systems of Linear Equations so the method and meaning are both clear.

Answer: (6, 4)

Practice problems

1. The graph solution to a system is the...

Choices: Intersection point · x-axis only · steepest line · largest y-intercept

Show solution
  1. Warm-up: First identify exactly what the question is asking: The graph solution to a system is the...
  2. For a system, use substitution, elimination, or graphing to find the value pair that makes both equations true.
  3. Both equations are true where their lines meet.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
  5. Identify the Algebra I structure before choosing a calculation.

Answer: Intersection point

2. Solve x + y = 8 and y = 3. Enter x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x + y = 8 and y = 3. Enter x.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute y = 3.
  4. x + 3 = 8, so x = 5.
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

3. Solve x = 2 and y = x + 5. Enter y.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x = 2 and y = x + 5. Enter y.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute x = 2.
  4. y = 2 + 5 = 7.
  5. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

4. Solve y = x + 1 and y = 5. Enter x.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve y = x + 1 and y = 5. Enter x.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Set x + 1 equal to 5.
  4. x = 4.
  5. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

5. Solve x + y = 12 and x = 7. Enter y.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve x + y = 12 and x = 7. Enter y.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute x = 7.
  4. 7 + y = 12, so y = 5.
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

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