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Chapter 8: Problem 58

Solve. $$ \frac{5}{13} y=130 $$

Short Answer

Expert verified
y = 338

Step by step solution

01

Isolate the variable

To solve for the variable y, first get rid of the fraction by isolating y. To do this, multiply both sides of the equation by the reciprocal of \( \frac{5}{13} \). The reciprocal of \( \frac{5}{13} \) is \( \frac{13}{5} \).
02

Multiply both sides by the reciprocal

Multiply both sides of the equation by \( \frac{13}{5} \): \[ \frac{13}{5} \times \frac{5}{13} y = \frac{13}{5} \times 130 \] This simplifies as: \[ y = \frac{13}{5} \times 130 \]
03

Simplify the right-hand side

Simplify the multiplication on the right-hand side: \[ y = \frac{13 \times 130}{5} \] First, multiply 13 and 130: \[ 13 \times 130 = 1690 \] Then, divide by 5: \[ y = \frac{1690}{5} = 338 \]

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Key Concepts

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

Fractional Coefficients
When solving linear equations, a fractional coefficient is a fraction that multiplies a variable. For example, in the equation \(\frac{5}{13} y = 130\)\, \(\frac{5}{13}\) is the fractional coefficient of y.
Dealing with fractions might seem tricky, but it鈥檚 about knowing how to manipulate them.
Here are some tips:
  • Treat the fraction just like any other number. You can add, subtract, multiply, and divide it.
  • To eliminate the fraction, you might want to use its reciprocal, which we'll talk about next.
Remember, once the fraction is gone, solving the equation becomes simpler.
Reciprocals
A reciprocal is basically a flipped fraction. If you have \(\frac{5}{13}\), the reciprocal is \(\frac{13}{5}\).
Why are reciprocals useful? They help eliminate fractions in equations.
Multiplying a fraction by its reciprocal results in 1. For example: \[\frac{5}{13} \times \frac{13}{5} = 1\] This trick simplifies the calculations and isolates the variable. In our exercise, we multiplied both sides of the equation by the reciprocal of \(\frac{5}{13}\) to make it simpler.
It looked like this: \(\frac{13}{5} \times \frac{5}{13} y = \frac{13}{5} \times 130\).
Isolating Variables
Isolating the variable means getting the variable (in this case, y) alone on one side of the equation.
After multiplying both sides by the reciprocal, we got \( y = \frac{13}{5} \times 130 \).
This makes the equation simpler and allows us to solve for y easily.
Here are the steps we followed to isolate the variable:
  • Found the reciprocal of the fractional coefficient.
  • Multiplied both sides of the equation by that reciprocal.
  • Simplified the resulting equation.
  • Solved for the final value of y.
By following these steps, you can isolate any variable and solve similar linear equations!

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