Problem 25 - Intermediate Algebra

Question

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{c} 4 x-3 y=2 \\ 5 x-y=3 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{7}{11}, y = \frac{2}{11} \).

Step 1: Choose the Method

For this system of equations, the elimination-by-addition method seems more efficient because the coefficients of \( y \) are simple, which makes it easier to eliminate one variable. We will eliminate \( y \) by aligning the coefficients.

Step 2: Make Coefficients Equal

Multiply the second equation by 3 to make the coefficients of \( y \) in both equations equal.\[3(5x - y) = 3(3)\]which simplifies to:\[15x - 3y = 9\]

Step 3: Subtract Equations

Subtract the new equation from the first equation to eliminate \( y \):\[(4x - 3y) - (15x - 3y) = 2 - 9\]which simplifies to:\[-11x = -7\]

Step 4: Solve for x

Solve for \( x \) by dividing both sides by \(-11\):\[x = \frac{7}{11}\]

Step 5: Substitute to Find y

Substitute \( x = \frac{7}{11} \) back into the second original equation:\[5\left(\frac{7}{11}\right) - y = 3\]Simplify and solve for \( y \):\[y = \frac{35}{11} - 3 = \frac{35}{11} - \frac{33}{11} = \frac{2}{11}\]

Key Concepts

Substitution Method

The substitution method is a useful way to find solutions in a system of linear equations, especially when equations are simple and easily manipulated. This approach involves expressing one variable in terms of the other, then substituting this expression into the other equation. Through substitution, we simplify a system with two variables into a single equation with one variable.
Imagine you have two equations:

• Equation 1: \( y = 2x + 3 \)
• Equation 2: \( x + y = 10 \)

To use the substitution method, start by solving Equation 1 for \( y \), which is already done here. Next, substitute \( y = 2x + 3 \) into Equation 2:

• \( x + (2x + 3) = 10 \)
• Simplify to: \( 3x + 3 = 10 \)
• Solve for \( x \): \( 3x = 7 \), giving us \( x = \frac{7}{3} \)

Now substitute \( x = \frac{7}{3} \) back into Equation 1:

• \( y = 2\left(\frac{7}{3}\right) + 3 \)
• Simplify to find \( y \).

This method is ideal when one equation is easily solvable for one of the variables. It is straightforward and breaks down complex systems into manageable steps.

Elimination Method

The elimination method, also known as the addition or subtraction method, is another effective way to solve systems of linear equations. This approach aims to eliminate one of the variables by adding or subtracting the equations, allowing you to solve the remaining variable.
Let's look at a system of equations:

• Equation 1: \( 2x + 3y = 6 \)
• Equation 2: \( 4x - 3y = 12 \)

To eliminate \( y \), add both equations:

• \( (2x + 3y) + (4x - 3y) = 6 + 12 \)
• The \( y \) terms cancel: \( 6x = 18 \)
• Solve for \( x \): \( x = 3 \)

Next, substitute \( x = 3 \) back into either original equation to solve for \( y \). Let's use Equation 1:

• \( 2(3) + 3y = 6 \)
• \( 6 + 3y = 6 \), leading to \( y = 0 \)

The elimination method is powerful and works well when the coefficients of one of the variables can easily be made equal. This method provides a clear and straightforward way to simplify and solve equations.

Solving Linear Equations

Solving linear equations involves finding the values of the variables that make the equation true. A linear equation is an equation where each term is either a constant or the product of a constant and a single variable. Linear equations can be visualized as straight lines when graphed on a coordinate plane.
The standard form of a linear equation in two variables is \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. Solving such equations typically involves one or more of the following techniques:

• Isolating the variable: Performed by adding, subtracting, multiplying, or dividing both sides of the equation by a constant.
• Substitution or elimination, as explained, for systems of equations: These methods help in solving the system by reducing multiple equations to one-variable equations.

Consider a simple equation:

• \( 3x + 4 = 19 \)
• Isolate \( x \): Subtract 4 from both sides, \( 3x = 15 \)
• Divide by 3, \( x = 5 \)

These basic steps are essential foundation skills in algebra. Mastering them allows for further progression into more complex algebra and realism in mathematical modeling.