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Solve the Linear Equation -5(x - 4) = -15 by First Dividing Both Sides
Mathematics
Grade 7 (Junior High School)
Question Content
Solve the equation. What is the value of x? One way you could solve this equation is by dividing first. Show how you could divide both sides by the same number so that one group of (x - 4) remains on the left. Then, rewrite the equation. The equation is: -5(x - 4) = -15
Correct Answer
x = 1; The filled blanks are: -5, -5, (x - 4), 3
Detailed Solution Steps
1
Step 1: Identify the coefficient of the group (x - 4) on the left side of the equation, which is -5. To leave only (x - 4) on the left, we need to divide both sides of the equation by this coefficient, -5.
2
Step 2: Perform the division on the left side: $\frac{-5(x - 4)}{-5} = (x - 4)$
3
Step 3: Perform the division on the right side: $\frac{-15}{-5} = 3$
4
Step 4: Rewrite the simplified equation: $x - 4 = 3$
5
Step 5: Solve for x by adding 4 to both sides: $x = 3 + 4 = 7$
Knowledge Points Involved
1
Properties of Equality (Division Property)
This property states that if you divide both sides of a true equation by the same non-zero number, the resulting equation is still true. It is used to simplify equations and isolate the variable term, which is essential for solving linear equations like the one in this problem.
2
Solving One-Variable Linear Equations
One-variable linear equations are equations in the form $ax + b = c$ (or variations with grouped terms). The goal is to isolate the variable by using inverse operations (like division, addition/subtraction) to find its value. This problem uses division first to simplify the grouped term before isolating x.
3
Division of Negative Numbers
When dividing two negative numbers, the result is a positive number. For example, $\frac{-15}{-5} = 3$. This rule applies to all real numbers and is critical for solving equations with negative coefficients or constants.
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