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Chapter 4: Problem 87

For exercises \(87-90\), solve by graphing. Sketch the graph; describe the window; identify the solution. $$ \begin{aligned} &y=x+5 \\ &y=3 x+9 \end{aligned} $$

Short Answer

Expert verified
The graphs intersect at (2, 8).

Step by step solution

01

- Create Table of Values

First, create a table of values for both equations. Choose at least two values for x and compute the corresponding y-values for each equation. For example, for the first equation:\( y = 3x + 2 \)If \( x = 0 \), then \( y = 3(0) + 2 = 2 \). If \( x = 1 \), then \( y = 3(1) + 2 = 5 \).For the second equation:\( y = x + 6 \)If \( x = 0 \), then \( y = 6 \). If \( x = 1 \), then \( y = 7 \).
02

- Plot Points on Graph

Use the values obtained from the table to plot points on the graph for both equations. For example, plot (0, 2) and (1, 5) for the first equation, and plot (0, 6) and (1, 7) for the second equation.
03

- Draw Lines

Connect the points plotted for each equation using a straight edge to draw the lines representing both equations. Ensure the lines extend through the plotted points for accuracy.
04

- Identify Intersection

Observe where the two lines intersect on the graph. The coordinates of the intersection point represent the solution to the system of equations. Note this point down.
05

- Describe the Window

Describe the viewing window used to clearly see the intersection. Ensure the window includes the x-values and y-values plotted. A recommended viewing window could span from \(-5 \) to \(5 \) for both x and y axes.
06

- Verify the Solution

Optionally, verify the solution by substituting the intersection point back into the original equations to confirm both equations hold true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations form straight lines when graphed on a coordinate plane. They typically appear in the form: \( y = mx + b \), where \(m\) is the slope (the line's steepness) and \(b\) is the y-intercept (where the line crosses the y-axis). Linear equations are useful for showing a constant rate of change.

In our example, we have two linear equations:
  • \( y = 3x + 2 \)
  • \( y = x + 6 \)


Both equations will create straight lines. The difference in their slopes and y-intercepts will cause them to intersect at some point on the graph.
Graphing
Graphing is a visual way of solving systems of equations. By plotting points from each equation on the same graph and drawing the corresponding lines, we can find where they intersect. Follow these steps for effective graphing:

1. Choose values for x.
2. Calculate corresponding y-values.
3. Plot each \( (x, y) \) point on the coordinate plane.
4. Draw lines through the points for each equation.

Graphing helps us easily see the relationship between the equations and locate their intersection point.
Intersection Point
The intersection point on a graph represents the solution to the system of equations. It is the coordinate \( (x, y) \) where both lines cross, meaning both equations are true with these values.

In our example:
  • Graph \( y = 3x + 2 \)
  • Graph \( y = x + 6 \)
  • Observe where the lines intersect


Once you find this intersection, you've found the solution to the system. Verify by substituting these values back into both original equations to ensure they hold true.
Table of Values
A table of values helps organize the points to plot on your graph. By choosing x-values and calculating the matching y-values, you get coordinates to draw the graph correctly.

For the equations:
  • \( y = 3x + 2 \): If x=0, then y=2; if x=1, then y=5.
  • \( y = x + 6 \): If x=0, then y=6; if x=1, then y=7.


Plot these points on the graph and use a straight edge to connect them. The resulting lines will show you the relationship between variables clearly and help locate the intersection.

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Most popular questions from this chapter

The costs to deliver a course at a college include $$\$ 2300$$ per section plus an additional $$\$ 20$$ in materials per student. The revenue from tuition is about $$\$ 175$$ per student. Find the number of students at which the cost of delivering a course is equal to the revenues from tuition, and find the revenues from tuition. Round \(u p\) to the nearest student, and round the tuition to the nearest hundred.

A truck leaves a town traveling at a constant speed of \(\frac{65 \mathrm{mi}}{1 \mathrm{hr}}\). After \(25 \mathrm{~min}\), a car follows the same route traveling at a constant speed of \(\frac{70 \mathrm{mi}}{1 \mathrm{hr}}\). Find the time in minutes when the car will catch up with the truck. Find the distance traveled by each vehicle. Round to the nearest whole number.

The cost to make a product is $$\$ 21.50$$. The fixed overhead costs per month to make the product are $$\$ 21,450$$. The price of each product is $$\$ 29.75$$. Find the break-even point for this product.

For exercises 55-58, (a) Write four inequalities that represent the constraints. (b) Graph the inequalities that represent the constraints. Label the feasible region. An investor will put a maximum of $$\$ 25,500$$ in foreign investments and domestic investments with at least four times as much in domestic investments as in foreign investments and a minimum of $$\$ 2000$$ in foreign investments. Let \(x=\) amount in foreign investments, and let \(y=\) amount of domestic investments.

A tugboat leaves a port on the Columbia River at a speed of \(10 \mathrm{mi}\) per hour. One hour later, a powerboat leaves the port traveling at a speed of 24 mi per hour. Find the time and distance traveled by the powerboat when it catches up with the barge. Round to the nearest minute.
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