Problem 51 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{array}{l} \frac{x}{2}+\frac{y}{3}=1 \\ \frac{x}{4}-y=11 \end{array}\right.$$

Step-by-Step Solution

Answer

x = 8, y = -9

Step 1 - Clear Fractions in First Equation

First, clear the fractions in the first equation. Multiply every term in the equation \( \frac{x}{2} + \frac{y}{3} = 1 \) by 6 (the least common multiple of 2 and 3) to get: \[ 3x + 2y = 6 \]

Step 2 - Clear Fractions in Second Equation

Next, clear the fraction in the second equation. Multiply every term in \( \frac{x}{4} - y = 11 \) by 4 to get: \[ x - 4y = 44 \]

Step 3 - Solve for x in One Equation

We solve for one variable. In this instance, we will start with the second equation \( x - 4y = 44 \). Solving for x gives us: \[ x = 44 + 4y \]

Step 4 - Substitute into the Other Equation

Substitute \( x = 44 + 4y \) in the first equation \( 3x + 2y = 6 \): \[ 3(44 + 4y) + 2y = 6 \] \[ 132 + 12y + 2y = 6 \] \[ 132 + 14y = 6 \]

Step 5 - Solve for y

Solve the equation from Step 4 for y: \[ 132 + 14y = 6 \] \[ 14y = 6 - 132 \] \[ 14y = -126 \] \[ y = \frac{-126}{14} \] \[ y = -9 \]

Step 6 - Solve for x using y value

Substitute \( y = -9 \) back into the equation \( x = 44 + 4y \): \[ x = 44 + 4(-9) \] \[ x = 44 - 36 \] \[ x = 8 \]

Step 7 - Verify the Solution

Substitute \( x = 8 \) and \( y = -9 \) back into the original equations to make sure both hold true: \(\frac{8}{2} + \frac{-9}{3} = 1 \) \[ 4 - 3 = 1 \] (True), and \(\frac{8}{4} - (-9) = 11 \) \[ 2 + 9 = 11 \] (True). Hence, the solution is verified.

Key Concepts

Linear Equations

Linear equations are essential in algebra and appear in various forms. In our exercise, we are given a system with two linear equations. The general form of a linear equation in two variables is: \[ ax + by = c \] where \( a \), \( b \), and \( c \) are constants. These equations graph as straight lines on a coordinate plane.

The primary skill required is understanding how to manipulate and solve these equations to find the pair \( (x, y) \) that satisfies both equations simultaneously.

Substitution Method

The substitution method is a technique for solving linear systems. Here's how it works:

• First, solve one of the equations for one variable.
• Next, substitute this expression into the other equation.
• Solve the substituted equation for the remaining variable.

In the provided solution, we first solved the second equation for \( x \) to get \( x = 44 + 4y \).

Then, we substituted \( x \) into the first equation to form an equation in terms of \( y \). This step is crucial as it simplifies the problem, making it easier to find the solution for one variable before finding the other.

Fractions in Equations

Dealing with fractions in linear equations can be tricky. They often complicate the process, making multiplication a valuable tool to clear them out.

In our exercise, both equations had fractions: \( \frac{x}{2} + \frac{y}{3} = 1 \) and \( \frac{x}{4} - y = 11 \). To remove the fractions:

• We multiplied the entire first equation by 6, which is the least common multiple (LCM) of 2 and 3, to simplify to \( 3x + 2y = 6 \).
• Next, we multiplied the second equation by 4 to get \( x - 4y = 44 \).

These steps help to make the system easier to handle by working with whole numbers instead of fractions.

Solutions Verification

Verifying solutions is vital to ensure that the values obtained indeed solve the system of equations. To verify:

• Substitute the values back into the original equations.
• Check if both equations hold true with these values.

In our worked example:

• For the first equation: \( \frac{8}{2} + \frac{-9}{3} = 1 \), verifying to \( 4 - 3 = 1 \), which is true.
• For the second equation: \( \frac{8}{4} - (-9) = 11 \), verifying to \( 2 + 9 = 11 \), also true.

Since both equations are satisfied, our solution \( x = 8 \) and \( y = -9 \) is confirmed. This is a crucial step because errors could arise, and verification ensures accuracy.