Problem 39 - Intermediate Algebra

Question

Use a graphing calculator to solve each system. Give all answers to the nearest hundredth. See Using Your Calculator: Solving Systems by Graphing. $$ \left\\{\begin{array}{l} 1.7 x+2.3 y=3.2 \\ y=0.25 x+8.95 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = -8.30\), \(y = 6.88\).

Step 1: Understanding the Equations

We have two equations to solve: (1) \(1.7x + 2.3y = 3.2\) and (2) \(y = 0.25x + 8.95\). The goal is to find the values of \(x\) and \(y\) where these lines intersect.

Step 2: Set Up the Equations on a Graphing Calculator

We should input the two equations into the calculator. The first equation can be rearranged into \(y = \frac{-1.7}{2.3}x + \frac{3.2}{2.3}\), while the second equation is already in \(y = mx + b\) form.

Step 3: Rearrange Equation 1 in Slope-Intercept Form

Convert the first equation into the form \(y = mx + b\) for graphing: \(2.3y = -1.7x + 3.2\). Divide each term by 2.3: \(y = -\frac{1.7}{2.3}x + \frac{3.2}{2.3}\). Calculate the values: \(y = -0.7391x + 1.3913\).

Step 4: Graph the Equations

Enter both equations into the graphing calculator: \(y = -0.7391x + 1.3913\) and \(y = 0.25x + 8.95\). Use the calculator to plot these lines and visually determine their point of intersection.

Step 5: Find the Point of Intersection

Use the 'intersect' function on the graphing calculator to find the coordinates where the two lines meet. Adjust viewing windows if necessary to ensure the intersection is visible.

Step 6: Record Intersection Coordinates

The intersection point given by the graphing calculator is \((x, y) = (-8.30, 6.88)\). Round these to the nearest hundredth if needed: \(x = -8.30\), \(y = 6.88\).

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool used in solving systems of equations by visualizing their solutions. It allows us to input equations and graph them visually. Many graphing calculators have functions like 'intersect' to easily find where lines cross.
When solving a system of equations, plotting the equations helps us see where they meet. This is especially helpful with complex numbers or decimals. Be sure to adjust the graph view if you can't see an intersection. Try zooming in or out.
Features to use include:

• Graph plotting: Enter equations like \(y = mx + b\) to see their lines on the graph.
• Intersect function: Find the exact point where two lines meet.
• Zoom and pan: Adjust your view to spot intersections easily.

This tool helps you to solve equations accurately and reinforces understanding by visual demonstration.

Slope-Intercept Form

The slope-intercept form of an equation is very useful for graphing and solving. It looks like \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
This format tells us how steep a line is and where it crosses the y-axis, guiding us in graphing. To convert any linear equation to this form, isolate \(y\) on one side.

Example: Convert \(1.7x + 2.3y = 3.2\) to slope-intercept form:

• Start by isolating \(y\): \(2.3y = -1.7x + 3.2\)
• Divide each term by 2.3: \(y = -0.7391x + 1.3913\)

The transformed equation makes graphing straightforward. You directly see the slope and intercept values needed to plot, which helps in finding intersections when graphing systems of equations.

Point of Intersection

The point of intersection is where two lines meet on a graph. Finding it gives the solution to a system of equations. In a graphing calculator, use the 'intersect' function to pinpoint this exact spot.
Plot both equations to see them intersect visually. The coordinates where they cross are the solution to your equation system. For example, in our case, the coordinates \((-8.30, 6.88)\) come from the intersection of \(y = -0.7391x + 1.3913\) and \(y = 0.25x + 8.95\).
Seeing this graphically makes it easier to understand how equations relate and what their solutions represent. Alter the graph's scale if needed to clearly view the intersection point, ensuring it isn't clipped out of view. This hands-on approach deepens comprehension of the equations and their interactions.