8th Grade Mathematics — Algebra I — Patterns in God's Creation
Finding Truth Through Balance — One-Step and Two-Step Equations
An equation is a mathematical statement that two expressions are equal. It always contains an equals sign (=). For example, 3x + 5 = 17 is an equation. The goal of solving an equation is to find the value of the variable that makes the statement true.
Think of an equation as a balance scale. Whatever is on the left side must equal whatever is on the right side. To keep the scale balanced, any operation you perform on one side must also be performed on the other side. This principle of balance is fundamental to all of algebra.
One-step equations require just one operation to isolate the variable. To solve, use the inverse (opposite) operation. Addition and subtraction are inverse operations; multiplication and division are inverse operations.
Example 1: x + 7 = 15. To isolate x, subtract 7 from both sides: x + 7 - 7 = 15 - 7, so x = 8. Example 2: 4x = 28. To isolate x, divide both sides by 4: 4x ÷ 4 = 28 ÷ 4, so x = 7. Always check your answer by substituting it back into the original equation.
Two-step equations require two operations to solve. The general strategy is to first undo addition or subtraction, then undo multiplication or division. Work in reverse order of operations.
Example: 3x + 5 = 17. Step 1: Subtract 5 from both sides: 3x = 12. Step 2: Divide both sides by 3: x = 4. Check: 3(4) + 5 = 12 + 5 = 17. The solution is correct.
Example: (x/2) - 3 = 7. Step 1: Add 3 to both sides: x/2 = 10. Step 2: Multiply both sides by 2: x = 20. Check: 20/2 - 3 = 10 - 3 = 7. Correct.
Sometimes variables appear on both sides of an equation. To solve, first collect all variable terms on one side and all constants on the other side.
Example: 5x + 3 = 2x + 18. Step 1: Subtract 2x from both sides: 3x + 3 = 18. Step 2: Subtract 3 from both sides: 3x = 15. Step 3: Divide both sides by 3: x = 5. Check: 5(5) + 3 = 28 and 2(5) + 18 = 28. Both sides equal 28, confirming x = 5.
The process of solving equations teaches us an important principle: truth is discovered through careful, logical steps. Just as we cannot jump to conclusions in mathematics without following proper procedures, we should approach all truth-seeking — including Biblical study — with careful reasoning and faithfulness to the evidence.
Write thoughtful responses to the following questions. Use evidence from the lesson text, Scripture references, and primary sources to support your answers.
Solve the equation 4x - 9 = 23. Show all steps and check your answer.
Guidance: Add 9 to both sides to get 4x = 32, then divide both sides by 4 to get x = 8. Check: 4(8) - 9 = 32 - 9 = 23.
How is the concept of balance in equations similar to the balance and justice that God values (Proverbs 11:1)?
Guidance: Consider how equations require equal treatment of both sides, just as God values honest and balanced dealings. Mathematical balance reflects moral balance.
Solve: 7x + 2 = 3x + 22. Explain each step and why you perform each operation.
Guidance: Subtract 3x from both sides (4x + 2 = 22), subtract 2 from both sides (4x = 20), divide by 4 (x = 5). Each step uses an inverse operation to isolate the variable.