Linear Equations and Graphs Checkpoint
A free College Algebra lesson from the “Linear Equations and Inequalities” unit, with a worked example and practice problems including step-by-step solutions.
This checkpoint reviews the linear foundation for College Algebra: equations, inequalities, slope, line forms, and related lines.
What you'll learn
- Solve linear equations and inequalities
- Use slope and line forms
- Connect equations to graphs
Why it matters: This checkpoint shows whether a student can move between solving, graphing, and interpreting linear relationships. That foundation supports placement readiness because later College Algebra topics still rely on equations, intervals, slope, and line forms.
Worked example
Problem. Find the slope through (1, 4) and (5, 12).
- Change in y is 8.
- Change in x is 4.
- Slope is 8/4 = 2.
Answer: 2
Practice problems
1. Solve 4x - 9 = 23.
Show solution
- Equations: First identify exactly what the question is asking: Solve 4x - 9 = 23.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Add 9, then divide by 4.
- Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.
Answer: 8
2. Solve -3x < 18. Enter the boundary number.
Show solution
- Inequalities: First identify exactly what the question is asking: Solve -3x < 18. Enter the boundary number.
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- Divide by -3 and reverse the sign.
- Check the result by substituting or estimating: the response should match -6 and make sense in the original problem.
Answer: -6
3. Find the slope through (2, 7) and (8, 19).
Show solution
- Slope: First identify exactly what the question is asking: Find the slope through (2, 7) and (8, 19).
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- 12/6 = 2.
- Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.
Answer: 2
4. A line with slope 4 through (3, 5) can be written as...
Choices: y - 5 = 4(x - 3) · y - 3 = 4(x - 5) · y + 5 = 4(x + 3) · y = 4x + 5
Show solution
- Line Forms: First identify exactly what the question is asking: A line with slope 4 through (3, 5) can be written as...
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- Use point-slope form.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: y - 5 = 4(x - 3)
5. Solve 2(3x - 4) = 22.
Show solution
- Equations: First identify exactly what the question is asking: Solve 2(3x - 4) = 22.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Divide by 2, add 4, then divide by 3.
- 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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