Problem 121 - Introductory Algebra for College Students

Question

Exercises \(120-122\) will help you prepare for the material covered in the first section of the next chapter. In each exercise, determine whether the given number is a solution of the equation. $$5 y+3-4 y-8=15 ; 20$$

Step-by-Step Solution

Answer

Yes, the number 20 is indeed a solution to the equation.

Step 1 - Simplify the Equation

Before checking if the number is a solution, simplify the equation. The equation is \(5y + 3 - 4y - 8 = 15\). Combine like terms: \(5y - 4y + 3 - 8 = 15\) which simplifies further to \(y - 5 = 15\).

Step 2 - Solve for y

Solve the simplified equation for y. In this case, add 5 to both sides. Thus y = 15 + 5 which gives y = 20.

Step 3 - Check the Solution

The number given for checking is 20. Since y = 20 from our previous step, it's clear the number 20 is indeed a solution to the equation.

Key Concepts

Simplifying Equations

In algebra, simplifying an equation involves combining all like terms so that the equation becomes more manageable. Start by identifying terms that can be combined. This means looking for terms where the variables (like \( y \) in our example) are the same, and then performing the necessary arithmetic operations.
For instance, in the equation \( 5y + 3 - 4y - 8 = 15 \), the terms \( 5y \) and \( -4y \) are like terms because they both involve \( y \). Combine them by subtracting \( 4y \) from \( 5y \), resulting in \( y \).
Next, combine the constant terms, \( 3 \) and \( -8 \), which results in \(-5\). So, the original equation simplifies to \( y - 5 = 15 \). This step makes solving the equation easier by reducing complexity. Always remember to maintain the balance of the equation by performing operations equally on both sides.

Solving Linear Equations

Once the equation is simplified, the aim is to isolate the variable and find its value. This often means performing arithmetic operations that "undo" each other until the variable is by itself on one side of the equation.
In our simplified equation \( y - 5 = 15 \), we need \( y \) alone on one side. Add \( 5 \) to both sides of the equation, as it cancels out the \( -5 \) on the left, leading to \( y = 20 \).
Each step should maintain equality, meaning whatever you do to one side, you must do to the other. This ensures the equation remains true as you simplify and solve it. Solving linear equations generally requires two straightforward steps: simplifying first, followed by isolating the variable by undoing any additions, subtractions, multiplications, or divisions.

Checking Solutions

The final step in solving an algebra equation should always be to check your solution. This step helps confirm the correctness of your answer by substituting the value back into the original equation.
For example, in our case, we need to verify if \( y = 20 \) is a valid solution. Substitute \( 20 \) back into the original equation \( 5y + 3 - 4y - 8 = 15 \). This gives \( 5(20) + 3 - 4(20) - 8 \).
Calculate each term: \( 100 + 3 - 80 - 8 \). This simplifies to \( 15 = 15 \), which verifies our solution because both sides of the equation are equal. This checking process is crucial to ensure the initial steps performed were correct and help avoid errors in more complex problems.