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Chapter 1: Problem 92

Solve the equation. $$ \frac{4}{5 z-3}=\frac{-2}{4 z+7} $$

Short Answer

Expert verified
The solution is \(z = -\frac{11}{13}\).

Step by step solution

01

Cross-multiply

Multiply both sides of the equation by the denominators to eliminate them. This gives: \(4(4z + 7) = -2(5z - 3)\)
02

Distribute

Distribute both sides: \(16z + 28 = -10z + 6\)
03

Combine like terms

Move all the terms involving \(z\) to one side and constant terms to the other side: \(16z + 10z = 6 - 28\)
04

Simplify

Combine the terms: \(26z = -22\)
05

Solve for z

Divide both sides by 26 to isolate \(z\): \(z = \frac{-22}{26} = -\frac{11}{13}\)

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Key Concepts

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

cross-multiplication
When solving rational equations, a common technique is cross-multiplication. This method helps eliminate the fractions by creating a single equation without denominators. Consider the equation: \( \frac{4}{5z - 3} = \frac{-2}{4z + 7} \).
To cross-multiply:
  • We multiply the numerator of the left-hand side by the denominator of the right-hand side.
  • Similarly, we multiply the numerator of the right-hand side by the denominator of the left-hand side.
This transforms the equation into: \[4(4z + 7) = -2(5z - 3)\].
No more fractions to deal with, making it much easier to solve.
distributive property
Next, we apply the distributive property to simplify the equation we obtained from cross-multiplication. The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\).
For our equation, we apply this property to both sides:
  • Distribute 4 to both terms inside the parenthesis: \(4(4z + 7)\) becomes \(16z + 28\).
  • Distribute -2 to both terms inside the parentheses: \(-2(5z - 3)\) becomes \(-10z + 6\).
After distributing, our equation looks like this: \[16z + 28 = -10z + 6\].
This simplification makes it easier to isolate the variable in the next steps.
isolating variable
Finally, we want to isolate the variable \(z\) to find its value. Starting from the equation \(16z + 28 = -10z + 6\):
  • First, move all terms involving \(z\) to one side by adding \(10z\) to both sides: \(16z + 10z + 28 = 6\).
  • Combine the \(z\) terms: \(26z + 28 = 6\).
  • Next, isolate \(z\) by subtracting 28 from both sides: \(26z = -22\).
  • Finally, divide both sides by 26: \(z = \frac{-22}{26} = -\frac{11}{13}\).
By isolating \(z\), we determine that \(z = -\frac{11}{13}\).
This method of rearranging and combining like terms is crucial for solving any equation.

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Most popular questions from this chapter

Suppose that \(P\) dollars in principal is invested at an annual simple interest rate \(r\) for \(t\) years. Then the amount in the account \(A\) (in \$) is given by \(A=P+P r t\). With a \(4 \%\) simple interest rate, what is the minimum amount of principal required for an investment to grow to at least \(\$ 10,000\) in 8 yr? Round to the nearest cent.

Solve the equations. \(\left|4-\frac{1}{2} w\right|-\frac{1}{3}=\frac{1}{2}\)

Write an absolute value inequality that represents the statement. \(3 x-1<-7\) or \(3 x-1>7\)

Explain why the equation \(|x|=-5\) has no solution.

Explain the difference between the solution sets for the following inequalities: $$|x-3| \leq 0 \text { and }|x-3|>0$$
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