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Unit 1: Linear Relationships and Equations

Topic 1

Variables, Expressions, and Equations

Mathematical expressions and equations are the building blocks of algebra. Variables represent unknown values. We often use letters like xx, yy, or nn to stand for a number that can change.
An expression is a combination of numbers, variables, and operations, but it does not include an equals sign.
3x+5or2(a−4)3x + 5 \quad \text{or} \quad 2(a - 4)
An equation shows that two expressions are equal.
3x+5=173x + 5 = 17
To solve an equation means to find the value of the variable that makes the equation true. The goal is to isolate the variable on one side of the equation using opposite operations.

Core Skills

  • Simplify expressions by combining like terms and using the distributive property.
  • Translate word phrases into algebraic expressions.
  • Solve simple linear equations by isolating the variable.

Example 1: Simplifying an Expression

Simplify the expression:
3(2y−4)+y3(2y - 4) + y
Step 1: Apply the distributive property.
3×2y=6y,3×(−4)=−123 \times 2y = 6y, \quad 3 \times (-4) = -12
So the expression becomes:
6y−12+y6y - 12 + y
Step 2: Combine like terms.
6y+y=7y6y + y = 7y
Final Answer:
7y−12\boxed{7y - 12}

Example 2: Solving a Simple Equation

Solve for xx:
2x+5=192x + 5 = 19
Step 1: Subtract 5 from both sides.
2x+5−5=19−52x + 5 - 5 = 19 - 5
2x=142x = 14
Step 2: Divide both sides by 2.
x=142x = \frac{14}{2}
x=7\boxed{x = 7}
Check: Substitute x=7x = 7 back into the original equation:
2(7)+5=19✓2(7) + 5 = 19 \quad \checkmark

Key Takeaways

  • Expressions do not have an equals sign; equations do.
  • Always perform the same operation on both sides of an equation to keep it balanced.
  • Checking your solution helps confirm accuracy.