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Chapter 3: Problem 61

Solve each equation. \(50 y=7(7 y+4)\)

Short Answer

Expert verified
The solution to the equation is \( y = 28 \).

Step by step solution

01

Distribute on the Right Side

First, distribute the 7 on the right side of the equation into the expression inside the parentheses. This involves multiplying 7 by each term inside: \[ 50y = 7(7y + 4) \] which simplifies to: \[ 50y = 49y + 28 \]
02

Subtract 49y from Both Sides

Next, to isolate the variable term, subtract 49y from both sides of the equation: \[ 50y - 49y = 49y + 28 - 49y \] which simplifies to: \[ y = 28 \]
03

Verify the Solution

Substitute the solution back into the original equation to ensure it is correct. Substitute \( y = 28 \) into \( 50y = 7(7y + 4) \): \[ 50 imes 28 = 7(7 imes 28 + 4) \] Calculate both sides: \[ 1400 = 7(196 + 4) \] \[ 1400 = 7 imes 200 \] \[ 1400 = 1400 \] Both sides equal, confirming that \( y = 28 \) is the correct solution.

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Key Concepts

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

Understanding the Distributive Property
The distributive property is a fundamental concept in algebra that helps to simplify expressions and solve equations efficiently. It involves multiplying a single term by two or more terms inside parentheses. In simple terms, the distributive property allows you to 'distribute' the multiplication over addition or subtraction inside the parentheses.
For example, when you have an equation like \(7(7y + 4)\), you can use the distributive property to multiply 7 by each term within the parentheses:
  • Multiply 7 by \(7y\) to get \(49y\)
  • Then, multiply 7 by 4 to get 28
This simplification leads to the expression \(49y + 28\).
Applying the distributive property typically makes problem-solving easier and creates a clearer path towards isolating the variable.
Mastering Variable Isolation
Variable isolation is a crucial step in solving linear equations, enabling you to find the value of the unknown variable. The primary goal is to get the variable on one side of the equation all by itself.
In our problem, after applying the distributive property, we have the equation \(50y = 49y + 28\).
To isolate \(y\), we need to eliminate any terms involving \(y\) from one side. Here, you can subtract \(49y\) from both sides:
  • This results in \(50y - 49y = 28\), simplifying to \(y = 28\)
This process efficiently gets the variable on its own, allowing you to solve for its value. Remember, the key step is to perform the same operation on both sides of the equation to maintain balance.
Executing Solution Verification
Solution verification is an essential step to ensure that your answer is correct. This involves substituting the value you found back into the original equation.
For instance, in our exercise, after finding \(y = 28\), we substitute it back into the original equation \(50y = 7(7y + 4)\) and calculate:
  • Substitute 28: \(50 \times 28 = 7(7 \times 28 + 4)\)
  • Perform calculations: \(1400 = 7(196 + 4)\)
  • Further simplify to: \(1400 = 7 \times 200 = 1400\)
Both sides of the equation are equal, proving that the solution \(y = 28\) is indeed valid.
Solution verification not only confirms your answer but also helps you practice critical thinking and precision in mathematics.

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