Problem 43 - Elementary Algebra Within Reach

Question

In Exercises \(43-48\), write the equations of the lines in slope-intercept form. What can you conclude about the number of solutions of the system? $$ \left\\{\begin{array}{r} 2 x-3 y=-12 \\ -8 x+12 y=-12 \end{array}\right. $$

Step-by-Step Solution

Answer

The equations have been transformed into the following slope-intercept forms: \(y = \frac{2}{3}x + 4\) and \(y = \frac{-2}{3}x + 1\). The system of equations has one unique solution, where these lines intersect.

Step 1: Write First Equation in slope-intercept form

Start by rearranging the first equation, \(2x - 3y = -12\), to the slope-intercept form. To do this, isolate \(y\). First, subtract \(2x\) from both sides to get \(-3y = -2x - 12\). Then, divide each side by \(-3\) to solve for \(y\), which gives us \(y = \frac{2}{3}x + 4\)

Step 2: Write Second Equation in slope-intercept form

Moving to the second equation, \(-8x + 12y = -12\), rearrange it into slope-intercept form. Subtract \(-8x\) from both sides to get \(12y = 8x - 12\). Divide each side by \(12\), we get \(y = \frac{-2}{3}x + 1\)

Step 3: Interpret The Result

Now that both equations are in slope-intercept form, notice that the two lines have differing slopes and y-intercepts. Therefore, the lines will intersect at a single point. Hence, there is one unique solution to this system of equations.

Key Concepts

Slope-Intercept Form

One of the most common ways to express a linear equation is the slope-intercept form. This form is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept.
The slope indicates how steep the line is, while the y-intercept tells us where the line crosses the y-axis. To convert a linear equation to this form, you should solve for \( y \).
This involves isolating \( y \) on one side of the equation, often through basic operations such as addition, subtraction, multiplication, or division. Knowing how to do this provides a powerful tool for analyzing and graphing linear equations. In our exercise, this step reveals the characteristics of each line clearly, as seen with the rearrangements and solutions for \( y \).

• Identify the slope \( m \), which is the coefficient of \( x \).
• Locate the y-intercept \( b\), which is the constant term without \( x \).

Both equations can now be easily graphed and compared once they are in slope-intercept form.

Unique Solution

When two linear equations are transformed into the slope-intercept form, as seen in our exercise, analyzing them becomes straightforward. If the slopes of the two lines \( m_1\) and \( m_2\) are different, the lines intersect at exactly one point.
This intersection point is the unique solution to the system of equations. It's like finding the exact spot where two different paths cross.
In our exercise, the slopes clearly differ: one equation is \( y = \frac{2}{3}x + 4 \) and the other is \( y = \frac{-2}{3}x + 1 \).

• The slope \( \frac{2}{3} \) means for every 3 units moved horizontally, the line moves 2 units vertically.
• The slope \( \frac{-2}{3} \) means for every 3 units moved horizontally, the line moves 2 units downward.

Since they are not parallel, they will intersect, producing exactly one solution. This confirms the presence of a unique solution.

Linear Equations

Linear equations are a fundamental concept in algebra. They describe lines on a graph, and their general form can be captured in equations like \( Ax + By = C \). Each equation corresponds to a straight line when graphed.
The key components of linear equations are: the variables \( x \) and \( y \), and the constants \( A \), \( B \), and \( C \). In our context, we have two linear equations representing two lines.
The process of converting these equations to slope-intercept form reveals more about their nature compared to their original form.

• Linear equations can represent parallel lines, which have no solutions.
• Lines that intersect once provide a unique solution.
• Coincident lines, sharing the same path, have infinite solutions.

Understanding linear equations helps recognize these outcomes. By comparing slopes in slope-intercept form, we determine how these lines relate to each other, verifying the number of solutions.