Introduction
Solving Linear Equations in One Variable is an important Grade 8 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with solving linear equations in one variable.
What Is Solving Linear Equations in One Variable?
Solving Linear Equations in One Variable means choosing a model, naming what each number means, and explaining the strategy.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Solving Linear Equations in One Variable
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
Visual Model 1
Question: Look at the table of values for the equation \(y=2x+1\): What value of \(x\) produces \(y=9\)?
| \(x\) | \(y\) |
|---|---|
| 1 | 3 |
| 2 | 5 |
| \(?\) | 9 |
- A. \(x=3\)
- B. \(x=5\)
- C. \(x=8\)
- D. \(x=4\)
Why it works: Use \(y=2x+1\): \(9=2x+1 \Rightarrow 8=2x \Rightarrow x=4\).
Answer: \(x=4\)
Worked Examples
Example 1
Question: Solve for \(x\): \(3(x-4)=2x+5\)
- A. \(x=-7\)
- B. \(x=7\)
- C. \(x=17\)
- D. \(x=9\)
- Distribute: \(3x-12=2x+5\).
- Subtract \(2x\): \(x-12=5\).
- Add \(12\): \(x=17\).
- Check: \(3(17-4)=3(13)=39\) and \(2(17)+5=39\) ✓.
Answer: \(x=17\)
Example 2
Question: Solve for \(x\): \(2x+8=20\)
- A. \(x=6\)
- B. \(x=14\)
- C. \(x=28\)
- D. \(x=4\)
- Subtract \(8\) from both sides: \(2x=12\).
- Divide by \(2\): \(x=6\).
- Check: \(2(6)+8=12+8=20\) ✓.
Answer: \(x=6\)
Example 3
Question: Solve for \(x\): \(5x-3=2x+9\)
- A. \(x=2\)
- B. \(x=12\)
- C. \(x=6\)
- D. \(x=4\)
- Subtract \(2x\) and add \(3\): \(3x=12\).
- Divide by \(3\): \(x=4\).
- (Distractor A: result of dividing 12 by \(3\) incorrectly; C: forgetting to distribute constant; D: result of incorrect arithmetic.)
Answer: \(x=4\)
Real-World Word Problems
Problem 1
Question: Two students solve \(2x+5=11\) and get different answers: Student A gets \(x=3\) and Student B gets \(x=4\). Without solving, how can you determine which student is correct?
- A. Substitute each value into the original equation to check which one works
- B. Solve the equation and see who matched your answer
- C. Check which value is smaller
- D. Divide 11 by 2 to see which is closer
Why it works: Substitution is the definitive test: \(2(3)+5=11\) ✓ and \(2(4)+5=13 \neq 11\) ✗. Student A is correct.
Answer: Substitute to verify
Problem 2
Question: A plumber charges a \($50\) service call fee plus \($40\) per hour. If the total bill is \($170\), which equation represents the time worked (in hours)?
- A. \(50+40h=170\)
- B. \(50h+40=170\)
- C. \(40+50h=170\)
- D. \(50 \times 40 = 170h\)
Why it works: Base fee is \($50\), hourly rate is \($40\) per hour, total is \($170\). Equation: \(50+40h=170\). Solving: \(40h=120 \Rightarrow h=3\) hours.
Answer: \(50+40h=170\)
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
Solve for \(y\): \(-4y+7=15\)
- A. \(y=-8\)
- B. \(y=8\)
- C. \(y=2\)
- D. \(y=-2\)
Question 2
What is the solution to \(6(x+2)=30\)?
- A. \(x=3\)
- B. \(x=5\)
- C. \(x=8\)
- D. \(x=-2\)
Question 3
Solve for \(x\): \(\frac{x}{3}+5=9\)
- A. \(x=12\)
- B. \(x=36\)
- C. \(x=8\)
- D. \(x=14\)
Question 4
If an equation simplifies to \(0=5\) after subtracting variables and constants from both sides, what does this tell you?
- A. The equation has exactly one solution
- B. The solution is \(x=0\)
- C. The equation has infinitely many solutions
- D. The equation has no solution
Question 5
Which is the solution to \(4(2x-1)=3x+9\)?
- A. \(x=1\)
- B. \(x=3\)
- C. \(x=\frac{13}{5}\)
- D. \(x=-5\)
Question 6
Two equations are given: Equation 1: \(2(x+3)=2x+6\) and Equation 2: \(2(x+3)=2x+5\). Which statement is true?
- A. Both have exactly one solution
- B. Equation 1 has one solution; Equation 2 has infinitely many
- C. Both have no solution
- D. Equation 1 has infinitely many solutions; Equation 2 has no solution
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(y=-2\)
Subtract \(7\): \(-4y=8\). Divide by \(-4\): \(y=-2\).
Question 2
Answer: \(x=3\)
Distribute: \(6x+12=30\). Subtract \(12\): \(6x=18\). Divide by \(6\): \(x=3\).
Question 3
Answer: \(x=12\)
Subtract \(5\): \(\frac{x}{3}=4\). Multiply by \(3\): \(x=12\).
Question 4
Answer: The equation has no solution
A false statement like \(0=5\) means no value of the variable satisfies the equation.
Question 5
Answer: \(x=\frac{13}{5}\)
Distribute: \(8x-4=3x+9\). Subtract \(3x\) from both sides and add \(4\) to both sides to get \(5x=13\), so \(x=\frac{13}{5}\).
Question 6
Answer: Infinitely many for Eq. 1; no solution for Eq. 2
Eq. 1: Expand \(2x+6=2x+6\) (identity, infinitely many). Eq. 2: Expand \(2x+6=2x+5 \Rightarrow 6=5\) (false, no solution).
Connection to Standards
This lesson supports Grade 8 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Solving Linear Equations in One Variable becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Understand the model before choosing the operation.
