Why this matters for FAST: Linear equations are foundational for algebra and heavily tested on FAST. Students must be able to solve equations with multiple steps, use the distributive property, and recognize when equations have no solution or infinitely many solutions.
Students often distribute only to the first term inside parentheses. Example: 2(x + 3) becomes 2x + 3 instead of 2x + 6.
"When you distribute, you're giving the multiplier to EVERY term inside the parentheses. Think of it like giving a gift to everyone at the party - no one gets left out! 2(x + 3) = 2 times x PLUS 2 times 3 = 2x + 6."
Students subtract from one side but add to the other, or forget to change signs when moving terms across the equals sign.
"The equals sign is like a balance scale - whatever you do to one side, you MUST do to the other to keep it balanced. If you subtract 3 from the left, you subtract 3 from the right. The operation is the SAME on both sides!"
Students don't understand what it means when variables cancel out, leaving 5 = 5 (infinite solutions) or 3 = 7 (no solution).
"If all variables disappear and you get a TRUE statement like 5 = 5, that means ANY number works - infinite solutions! If you get a FALSE statement like 3 = 7, NO number can make this true - no solution!"
Review one-step and two-step equations. "If 2x = 10, what is x?" (x = 5). "If 3x + 4 = 19, what is x?" (x = 5). Remind students of the balance scale model.
"For multi-step equations, we follow a clear order: First, simplify each side by distributing and combining like terms. Then, move all variables to one side. Finally, solve like a two-step equation."
The 4-Step Process
1. SIMPLIFY: Distribute and combine like terms on each side
2. COLLECT: Move variables to one side, constants to other
3. ISOLATE: Get the variable term alone
4. SOLVE: Divide to find the variable value
Solve: 5x + 3 = 2x + 15
Step 1: 5x + 3 = 2x + 15 (already simplified)
Step 2: 5x - 2x + 3 = 15 (subtract 2x from both sides)
3x + 3 = 15
Step 3: 3x = 15 - 3 = 12 (subtract 3 from both sides)
Step 4: x = 12 / 3 = 4
Check: 5(4) + 3 = 23 and 2(4) + 15 = 23 ✓
"Sometimes when we solve, all the variables cancel out! If we get a TRUE statement like 0 = 0 or 5 = 5, the equation has INFINITE solutions - every number works! If we get a FALSE statement like 3 = 8, there's NO solution - no number can make it true!"
Special Case Examples
Work through these together:
"Solve: 7x + 2 = 4x + 14"
Solution: 7x - 4x = 14 - 2 → 3x = 12 → x = 4
Check: 7(4) + 2 = 30 and 4(4) + 14 = 30 ✓
For struggling students: Use algebra tiles or balance scale manipulatives. Focus on two-step equations first, then gradually add complexity. Have students write each step and name the operation they're performing.
For advanced students: Challenge them with equations involving fractions (e.g., x/2 + 3 = x/4 + 5) or equations with nested parentheses. Have them create their own equations with specified solutions.
For home: Send Parent Activity sheet. Families can use everyday scenarios like comparing phone plans or calculating when two quantities will be equal.