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Unit 1: Linear Relationships and Equations

Topic 2

Solving Linear Equations

A linear equation is an equation where the variable has an exponent of 1. When written in one variable, it has the general form:
ax+b=cax + b = c
where aa, bb, and cc are constants.
The goal in solving a linear equation is to isolate the variable on one side of the equation. We do this by using the properties of equality:
  • You can add or subtract the same number from both sides.
  • You can multiply or divide both sides by the same nonzero number.
The solution is the value of the variable that makes the equation true.

Core Skills

  • Simplify each side of the equation before solving.
  • Use inverse operations to isolate the variable.
  • Check the solution by substituting it back into the original equation.

Example 1: One-Step Equation

Solve for xx:
x−7=12x - 7 = 12
Step 1: Add 7 to both sides.
x−7+7=12+7x - 7 + 7 = 12 + 7
x=19x = 19
Check: Substitute x=19x = 19 into the equation.
19−7=12✓19 - 7 = 12 \quad \checkmark
Final Answer: x=19\boxed{x = 19}

Example 2: Two-Step Equation

Solve for xx:
4x−5=114x - 5 = 11
Step 1: Add 5 to both sides.
4x=164x = 16
Step 2: Divide both sides by 4.
x=164x = \frac{16}{4}
x=4\boxed{x = 4}
Check: Substitute x=4x = 4 into the equation.
4(4)−5=16−5=11✓4(4) - 5 = 16 - 5 = 11 \quad \checkmark

Key Takeaways

  • Perform the same operation on both sides of the equation.
  • Simplify carefully before isolating the variable.
  • Always verify your solution.