Problem 12 - Intermediate Algebra

Question

For Problems \(1-26\), solve each system by using the substitution method. (Objective 1) $$ \left(\begin{array}{c} y=\frac{3}{4} x-5 \\ 5 x-4 y=9 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{11}{2} \), \( y = -\frac{73}{8} \).

Step 1: Identify the Equations

We have two equations in the system: the first equation is \( y = \frac{3}{4}x - 5 \) and the second equation is \( 5x - 4y = 9 \).

Step 2: Substitute for y

From the first equation, we already have \( y \) expressed in terms of \( x \). Substitute \( y = \frac{3}{4}x - 5 \) into the second equation: \( 5x - 4(\frac{3}{4}x - 5) = 9 \).

Step 3: Simplify After Substitution

Distribute the \( -4 \) in the second equation: \( 5x - 3x + 20 = 9 \). Simplifying gives: \( 2x + 20 = 9 \).

Step 4: Solve for x

Subtract 20 from both sides: \( 2x = 9 - 20 \). Simplify to get \( 2x = -11 \). Divide both sides by 2 to find \( x = -\frac{11}{2} \).

Step 5: Substitute to Find y

Use the expression for \( y \) from Step 1: \( y = \frac{3}{4}x - 5 \). Substitute \( x = -\frac{11}{2} \) into this expression: \( y = \frac{3}{4}(-\frac{11}{2}) - 5 \).

Step 6: Simplify to Find y

Calculate \( y = -\frac{33}{8} - 5 \). Convert 5 to \( \frac{40}{8} \), so \( y = -\frac{33}{8} - \frac{40}{8} \). Simplify to get \( y = -\frac{73}{8} \).

Key Concepts

Systems of Equations

A system of equations is a set of two or more equations that share the same variables. In other words, these equations are inter-related and must be solved together to find common values for the variables involved. For example, in the provided exercise, the variables are \( x \) and \( y \), and their values must satisfy both equations simultaneously.
To solve a system of equations, there are several methods available:

• Substitution Method: This involves expressing one variable in terms of the other using one equation and then substituting it into the other equation.
• Elimination Method: This technique involves adding or subtracting equations in order to cancel out one of the variables, making it possible to solve for the other.
• Graphical Method: You plot both equations on the same graph to find the point(s) of intersection, which represents the solution.

In our exercise, the substitution method is used, which is ideal when one of the equations is already solved for one variable like \( y = \frac{3}{4}x - 5 \). This method utilizes the expression for one variable to replace it in the other equation, making it possible to solve for one variable at a time.

Linear Equations

Linear equations are equations that form a straight line when graphed. They come in the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. In a two-variable linear equation, the highest power of each variable is one.
The system in this exercise consists of linear equations:

• The first equation \( y = \frac{3}{4}x - 5 \) is in slope-intercept form, which makes it very easy to express \( y \) in terms of \( x \).
• The second equation \( 5x - 4y = 9 \) is in standard form. This equation didn't explicitly solve for either variable initially, but it is integral to finding the solution when combined with the first equation.

Linear equations can intersect in several ways:

• At a single point, indicating one unique solution.
• Everywhere, indicating infinitely many solutions, typical of identical equations.
• Nowhere, implying the lines are parallel and there is no solution.

In this case, by using the substitution method, we found a unique intersection point, showing there is a single solution for the variables \( x \) and \( y \).

Step-by-Step Algebra Solutions

One of the most effective ways to understand algebra is through step-by-step solutions. By unraveling each step clearly, students can follow and understand the logic behind each operation. Let's walk through the process used in the provided solution:

First, we identified the equations we want to solve. By labeling \( y = \frac{3}{4}x - 5 \) as the easier equation to manipulate, it was used for substitution into the second equation.
We then replaced \( y \) in \( 5x - 4y = 9 \) using the expression from the first equation, effectively reducing the system to a single linear equation with one variable, \( x \).
After performing operations such as distribution and simplification, we solved for \( x \). Subsequent steps involved substituting back that value into the equation for \( y \) and solving, ultimately finding a numeric value for \( y \).
Each of these steps ensures the process is transparent and reinforces the logical flow of solving systems of equations through substitution. Structured step-by-step solutions are invaluable for mastering algebra, as they offer clarity and allow students to check their work against a reliable process.