Problem 61 - Introductory Algebra for College Students

Question

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l} \frac{3 x}{5}+\frac{4 y}{5}=1 \\ \frac{x}{4}-\frac{3 y}{8}=-1 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = -1\) and \(y = 2\). Always remember to check the solutions in the original equations to ensure they are correct.

Step 1: Eliminate Fractions

First, multiply both equations by enough to eliminate fractions. The first equation should be multiplied by 5 and the second, by 8. This gives: \[\begin{{align*}}3x + 4y &= 5 \2x - 3y &= -8 \\end{{align*}}\]

Step 2: Use the Addition/Subtraction Method

To eliminate one of the variables, you can multiply the first equation by 2 and the second by 3, then subtract the second from the first. \[\begin{{align*}}6x + 8y &= 10\6x - 9y &= -24 \\end{{align*}}\]Subtracting, we get:\[17y = 34 \\]

Step 3: Solve for \(y\)

Divide both sides by 17 to find the value of \(y\). This gives \(y = 2\).

Step 4: Substitute \(y\) into one of the original equations

Substitute \(y = 2\) into the first of the original equations and solve for \(x\). This gives:\[ 3x + 4(2) = 5 \3x = -3 \x = -1 \]

Step 5: Check the solution

Verify if the obtained values of \(x\) and \(y\) are valid for both of the original equations. Substitute \(x = -1\) and \(y = 2\) into both the original equations to verify

Key Concepts

Fraction Elimination

Fractions often make equations look more complicated than they are. To simplify the equations, we start by eliminating these fractions. The goal is to convert the equations into a form that is easier to work with. In our exercise, the given equations contain fractions:

• \( \frac{3x}{5} + \frac{4y}{5} = 1 \)
• \( \frac{x}{4} - \frac{3y}{8} = -1 \)

To eliminate the fractions, we need to find a common multiple for the denominators in each equation:- For the first equation, multiply each term by 5 to get rid of the denominators.- For the second equation, multiply each term by 8.
This gives us:

• \( 3x + 4y = 5 \)
• \( 2x - 3y = -8 \)

By eliminating fractions, our equations are now easier to manage and we can proceed to the next step.

Addition/Subtraction Method

The Addition/Subtraction Method is a nifty tool for solving systems of equations. Once the fractions are gone, we can focus on getting rid of one of the variables. By adjusting the equations, it's possible to cancel out a variable altogether. Here's how it works in our problem:- We have the two simplified equations:

• \( 3x + 4y = 5 \)
• \( 2x - 3y = -8 \)

To use the addition/subtraction method, modify the equations to prepare them for elimination:

• Multiply the first equation by 2
• Multiply the second equation by 3

This gives you:

• \( 6x + 8y = 10 \)
• \( 6x - 9y = -24 \)

Now, subtract the second equation from the first:

• \( 6x + 8y - (6x - 9y) = 10 - (-24) \)

This results in one single equation with \( y \):

• \( 17y = 34 \)

From here, you can easily solve for \( y \).

Substitution Method

The Substitution Method is another well-regarded approach in solving equations, especially after finding one variable's value. It's especially helpful when the equations are already simplified. Here’s how it works in our context:After finding \( y = 2 \), you substitute this value back into one of the original equations to find \( x \). Using the first simplified equation is usually convenient:\[ 3x + 4(2) = 5 \]Let's solve step-by-step:

• First, compute \( 4(2) \), which equals 8.
• Replace this back in the equation to get: \( 3x + 8 = 5 \).
• Subtract 8 from both sides to isolate the term with \( x \): \( 3x = -3 \).
• Finally, divide by 3 to solve for \( x \): \( x = -1 \).

Now you know the values of both variables from the system.

Verification of Solutions

Verifying your solutions is crucial to ensure everything checks out correctly. After computing \( x = -1 \) and \( y = 2 \), substitute these values back into both original equations to confirm the solutions are correct:
First equation: \( \frac{3(-1)}{5} + \frac{4(2)}{5} \)

• Compute: \( -\frac{3}{5} + \frac{8}{5} = 1 \)

The first equation checks out!
Second equation: \( \frac{-1}{4} - \frac{3(2)}{8} \)

• Compute: \( -\frac{1}{4} - \frac{6}{8} = -\frac{1}{4} - \frac{3}{4} = -1 \)

This confirms that both solutions satisfy the original equations.
By verifying, you ensure that your solutions are not only correct but also consistent, reinforcing the integrity of your calculations.