Problem 121 - Introductory and Intermediate Algebra for College Students 4th

Question

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 given number 20 is a solution to the equation

Step 1: Combine Like Terms

In the algebraic equation \(5y + 3 - 4y - 8 = 15\), we combine like terms first. The equation then simplifies to \(y - 5 = 15\).

Step 2: Rearrange The Equation

Next, rearrange the equation to determine the value of y. Add 5 to both sides of the equation. The equation therefore simplifies to \(y = 15 + 5\). The solution for y here would have been 20.

Step 3: Substitute and Verify

To verify if 20 is a solution to the equation, we substitute y with 20 in the original equation. After substitution, the equation becomes \(5(20) + 3 - 4(20) - 8 ?= 15\). Simplifying this gives \(100 + 3 - 80 - 8 ?= 15 \). Simplifying further leads to \(15 ?= 15\). Both sides of the equation are equal, so 20 is indeed a solution to the equation.

Key Concepts

Combining Like Terms

Combining like terms is an important step in solving algebraic equations. It helps to simplify the expression. Like terms are terms that have the same variable raised to the same power. For example, in the equation \(5y + 3 - 4y - 8 = 15\), the terms \(5y\) and \(-4y\) are like terms because they both contain the variable \(y\).
To combine them, you simply perform the operation indicated by their coefficients. Here, that means subtracting \(4y\) from \(5y\). This leaves you with \(y\). The constant terms \(3\) and \(-8\) are also combined by subtraction to simplify to \(-5\).
This simplification results in the equation \(y - 5 = 15\). Making the equation simpler allows us to solve it more easily in the next steps.

Solving Equations

Once the equation is simplified, solving it involves finding the value of the variable that makes the equation true. In this case, from \(y - 5 = 15\), we solve for \(y\) by isolating it on one side of the equation.
To do this, we can move the constant term to the other side by performing the opposite operation. Because we have \(-5\) on the left, we need to add \(5\) to both sides. This results in \(y = 15 + 5\).

• This gives us \(y = 20\), indicating the solution of the equation is 20.

Rearranging and performing operations like these are crucial in algebra to keep the equation balanced while revealing the value of the unknown.

Substitution Method

The substitution method is helpful to verify if a potential solution truly satisfies the equation. After solving for \(y\), we found that \(y = 20\). Next, we substitute \(20\) back into the original equation to check our work.
Start with the original equation, \(5y + 3 - 4y - 8 = 15\), substitute \(y\) with \(20\): \(5(20) + 3 - 4(20) - 8 ?= 15\).

• Perform the multiplication and addition/subtraction: \(100 + 3 - 80 - 8\).
• Simplify to get \(15\).
• Both sides equal 15, confirming that 20 is indeed a solution.

By substituting the solution back into the original equation, this method acts as a double-check, ensuring that the solution is correct and providing confidence in the answer.