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Chapter 9: Problem 11

Solve each equation. Check each solution. $$ \frac{y}{5}+\frac{y}{2}=7 $$

Short Answer

Expert verified
The solution for the equation is \(y = 10\).

Step by step solution

01

Remove fractions from the equation

Find a common denominator for 5 and 2, which is 10. Multiply each side of the equation by 10: \(10*\frac{y}{5}+10*\frac{y}{2}=7*10\). Simplifying, this gives: \(2y+5y=70\)
02

Simplify the equation

Combine like terms. This reduces the equation to: \(7y=70\).
03

Solve for y

Solving for y, divide each side by 7. This gives: \(y=10\).
04

Validate the result

Substitute y with 10 back into the equation and check whether the left-hand side (LHS) equals the right-hand side (RHS). LHS:\(\frac{10}{5}+\frac{10}{2}=2+5=7\), this is equal to RHS (7). Therefore, the solution is correct.

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Key Concepts

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

Fractions in Equations
Fractions in equations can be tricky at first, but they are easy to handle once you get the hang of it. When you have an equation with fractions like \( \frac{y}{5} + \frac{y}{2} = 7 \), the key goal is to eliminate the fractions. This makes the equation simpler to solve.

Here's how you do it:
  • Identify the denominators of the fractions. In our example, they are 5 and 2.
  • Find a common denominator, which is the smallest number that both denominators go into evenly. For 5 and 2, the common denominator is 10.
  • Multiply every term of the equation by this common denominator to clear the fractions. So, we multiply the whole equation by 10: \[ 10 \times \left( \frac{y}{5} \right) + 10 \times \left( \frac{y}{2} \right) = 7 \times 10 \]
  • This step simplifies the equation to: \( 2y + 5y = 70 \).
By clearing fractions, we have transformed the equation into a simple linear equation. This will make the next steps much easier.
Combining Like Terms
Once fractions are removed, the next step is to simplify the equation. This involves combining like terms. Like terms are terms that contain the same variable raised to the same power.

In the equation \( 2y + 5y = 70 \), we have two like terms: \( 2y \) and \( 5y \). These can be combined together as follows:
  • Add up the coefficients of \( y \) (which are 2 and 5). This gives a new term: \( (2 + 5)y = 7y \).
  • The equation then becomes \( 7y = 70 \).
Combining like terms simplifies the equation further, making it much easier to solve as we only have one term with \( y \) left in the equation.
Validating Solutions
After solving for the variable, it's important to validate your solution to ensure it's correct. This involves substituting your value back into the original equation to check if both sides are equal.

Consider our solution where \( y = 10 \). We substitute \( y \) back into the original equation \( \frac{y}{5} + \frac{y}{2} = 7 \):
  • Substitute 10 for \( y \): \( \frac{10}{5} + \frac{10}{2} \)
  • Calculate each fraction: \( 2 + 5 = 7 \)
  • Compare the left side to the right side of the original equation: both equal 7.
Because the equation holds true (left side equals right side), this confirms that the solution \( y = 10 \) is indeed correct. This step is crucial because it verifies that no mistakes have been made in the process of solving the equation.

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