Problem 44 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} 4 x=3 y+7 \\ 8 x-6 y=14 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions.

Step 1: Rewrite each equation in slope-intercept form

First, the equations need to be in the form of \(y = mx + b\): For the first equation, start with \(4x = 3y + 7\): Rewriting it:\[4x - 7 = 3y\]Now solve for \(y\):\[y = \frac{4}{3}x - \frac{7}{3}\]For the second equation, start with \(8x - 6y = 14\):Rewriting it:\[8x - 14 = 6y\]Now solve for \(y\):\[y = \frac{4}{3}x - \frac{7}{3}\]

Step 2: Graph the equations

Plot the lines on the graph using the slope-intercept form. Both lines are the same: \[y = \frac{4}{3}x - \frac{7}{3}\]Since both equations are equivalent, the graphs will lie on top of each other.

Step 3: Identify the solution through the graph

Since the two lines overlap completely, every point on the line is a solution to the system of equations. This means there are infinitely many solutions.

Key Concepts

graphing linear equations

Graphing linear equations helps us visually understand where solutions to systems of equations lie. To start, getting the equations in the right form is crucial. You want to use the slope-intercept form, written as \(y = mx + b\). This makes it easier to plot the lines.

Once in this form, you can easily find the y-intercept (\(b\)) and the slope (\(m\)). The y-intercept is where the line crosses the y-axis, and the slope tells you how steep the line is.

When you graph these equations on the same set of axes, you can immediately see where they intersect. The intersection points are solutions to the system. In this case, you will see that the lines overlap one another. This visual representation is key to understanding the nature of the solutions.

slope-intercept form

The slope-intercept form of a linear equation is \(y = mx + b\). This form is very user-friendly for graphing because:

• \(b\) is the y-intercept, so you know where to start plotting on the y-axis.
• \(m\) is the slope, which tells you how to rise and run from the y-intercept.

Let's say we have an equation \(4x = 3y + 7\). To convert it to slope-intercept form, solve for \(y\):

• Rewriting gives us \(4x - 7 = 3y\)
• Solving for \(y\) gives us \(y = \frac{4}{3}x - \frac{7}{3}\)

Notice how the equation is now \(y = \frac{4}{3}x - \frac{7}{3}\), showing that \(m = \frac{4}{3}\) and \(b = -\frac{7}{3}\). This conversion allows us to easily graph any linear equation by identifying the slope and intercept.

infinitely many solutions

When you graph two lines and they overlap completely, it means that every point on the line satisfies both equations. In simpler terms, both equations describe the same line.

Consider the system:
\(4x = 3y + 7\) and \(8x - 6y = 14\).
After converting both to slope-intercept form, you'll see:

• First equation: \(y = \frac{4}{3}x - \frac{7}{3}\)
• Second equation: \(y = \frac{4}{3}x - \frac{7}{3}\)

Notice how both equations are identical. This means they will graph to the same line, leading us to conclude there are infinitely many solutions because the lines do not just intersect at one or several points—they share every point.

This happens when the system of equations is dependent, indicating that one equation is a multiple of the other. Identical lines mean each point on the line satisfies both equations, showcasing the concept of infinitely many solutions.