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Chapter 2: Problem 99

Solve each equation. Then check the result. $$ \frac{h}{-40}=5 $$

Short Answer

Expert verified
The solution to the equation is \( h = -200 \).

Step by step solution

01

Identify the Equation

The given equation is \( \frac{h}{-40}=5 \). We need to solve for the variable \( h \).
02

Multiply Both Sides by -40

To isolate \( h \), multiply both sides of the equation by \(-40\) to eliminate the fraction. This yields: \( h = 5 \times (-40) \).
03

Calculate the Product

Now, calculate \( 5 \times (-40) \). This results in \( h = -200 \). Thus, \( h = -200 \) is the solution.
04

Verify the Solution

To ensure the solution is correct, substitute \( h = -200 \) back into the original equation. Compute \( \frac{-200}{-40} \). This simplifies to \( 5 \), which matches the right side of the equation. Therefore, \( h = -200 \) satisfies the original equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Linear Equations
A linear equation is a type of equation where the highest power of the variable is 1. In simpler terms, a linear equation is one that forms a straight line when graphed because each term is either a constant or the product of a constant and a single variable. Here's the basic form:
  • Standard form: \( ax + b = c \)
  • Slope-intercept form: \( y = mx + b \)
These equations are called 'linear' because they describe relationships that create straight lines when plotted on a graph.
In the given problem, the linear equation is \( \frac{h}{-40} = 5 \). Even though it initially appears as a fraction, it is still linear, since it involves algebraic expressions that don't rise above the first degree.
Fraction Elimination
Dealing with fractions in equations can be tricky, but eliminating them makes solving much simpler. The strategy is to eliminate the fraction by performing operations that clear the denominator. Here's how you do it step-by-step:
  • Identify the denominator of the fraction you want to clear.
  • Multiply every term in the equation by this denominator. This cancels out the denominator from the fraction.
  • Simplify the equation after the cancellation.
In our case of \( \frac{h}{-40} = 5 \), multiplying every term by \(-40\) eliminates the fraction, leaving \( h = 5 \times (-40) \). This process creates a simpler linear equation where the variable can be easily isolated and determined.
Verification of Solutions
Verification of solutions is a crucial part of solving equations. It ensures that the solution we've found truly satisfies the original equation. Here's some guidance on how to verify:
  • Substitute the solution back into the original equation in place of the variable.
  • Perform the calculations to simplify the equation.
  • Check that both sides of the equation are equal after substitution.
Let's apply this to our problem: After solving, we found \( h = -200 \). Substitute \(-200\) back into \( \frac{h}{-40}\) to check: \( \frac{-200}{-40} = 5 \). Simplifying the left side gives \( 5 \), which matches the right side. This confirms that our solution \( h = -200 \) is indeed correct.

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