Problem 133 - Introductory and Intermediate Algebra for College Students 4th

Question

Is \(x-4 y=14\) a true statement for \(x=12\) and \(y=1 ?\)

Step-by-Step Solution

Answer

No, \(x - 4y = 14\) is not a true statement for \(x = 12\) and \(y = 1\)

Step 1: Substituting the value of x

Substitute the given value of x into the equation. This gives us: \(12 - 4y = 14\)

Step 2: Substituting the value of y

Now, substitute the given value of y into the equation from Step 1. This gives us: \(12 - 4*1 = 12 - 4 = 8\)

Step 3: Checking if the equation holds true

We now compare the outcome on the left side (8) against the value on the right side of the equation (14). Since 8 is not equal to 14, the equation does not hold true for the given values of x and y.

Key Concepts

Solving Equations

To determine if a given set of values satisfies an equation, such as \( x - 4y = 14 \), the process begins by solving the equation. Solving equations means finding the value of variables that make the equation true. In our original exercise, the task is to verify whether the equation holds true for given values. Consider the equation as a balance, where both sides have to equal each other. If they do, the equation is satisfied. In this case, you're not solving for \( x \) or \( y \) like we often do but checking if the balance holds with given values. This means substituting the values into the equation and solving any resulting arithmetic.

Substitution Method

One powerful method to test if values satisfy an equation is the substitution method. It involves replacing a variable within an equation with a specific value. This technique is incredibly useful when you're provided with certain values for the variables. In the original step-by-step solution:

• First, the given value of \( x = 12 \) is substituted into the equation, which modifies \( x - 4y = 14 \) to \( 12 - 4y = 14 \).
• Next, the given value of \( y = 1 \) is substituted into the equation, further simplifying it to \( 12 - 4 \times 1 = 12 - 4 = 8 \).

By performing this substitution, we aim to verify the truth of the equation given the values of \( x \) and \( y \). It's a straightforward step-by-step method that effectively shows if both sides of the equation can equal each other.

Checking Solutions

After performing the substitution method, the next step is to check if the values make both sides of the equation equal. This process is called checking solutions.

• You calculate the expression on the left side after substitution, which in this case resulted in 8, and compare it to the right side of the original equation, which is 14.
• If both sides are equal, then the equation holds true for the given values, confirming a solution. If they are not equal, as seen here (8 ≠ 14), then the provided values do not satisfy the equation.

Checking solutions helps you confirm whether or not the proposed solution is valid, essentially giving a final stamp of approval or disapproval on the values being tested.