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Chapter 5: Problem 16

Solve each system of equations by sketching the graphs. Use the slope and the \(y\)-intercept or both intercepts. Estimate the result to the nearest 0.1 if necessary. $$\begin{aligned} &2 y=x-8\\\ &y=2 x+2 \end{aligned}$$

Short Answer

Expert verified
The solution to the system of equations is approximately \((-2, -2)\).

Step by step solution

01

Identify Equations

We are given two equations: \( 2y = x - 8 \) and \( y = 2x + 2 \). Our task is to solve this system graphically.
02

Express in Slope-Intercept Form

Convert both equations to the slope-intercept form, \( y = mx + b \). For the first equation, \( 2y = x - 8 \) becomes \( y = \frac{1}{2}x - 4 \) after dividing every term by 2. The second equation is already in this form: \( y = 2x + 2 \).
03

Identify Slope and Intercept

For the line \( y = \frac{1}{2}x - 4 \), the slope \( m = \frac{1}{2} \) and the \( y \)-intercept \( b = -4 \). For \( y = 2x + 2 \), the slope \( m = 2 \) and the \( y \)-intercept \( b = 2 \).
04

Sketch the First Graph

Begin by plotting the \( y \)-intercept of \( -4 \) on the \( y \)-axis for \( y = \frac{1}{2}x - 4 \). Since the slope is \( \frac{1}{2} \), rise 1 unit and run 2 units to the right to place another point.
05

Sketch the Second Graph

Plot the \( y \)-intercept of \( 2 \) for the line \( y = 2x + 2 \). With a slope of 2, rise 2 units and run 1 unit right to find another point on this line. Connect these points to draw the line.
06

Find the Intersection

Look for the intersection point of the two lines drawn. This point will be the solution to the system of equations. Estimate the coordinates by observing where the lines intersect on the graph.
07

Estimate the Intersection

The intersection point of the graphs appears around \( x = -2 \) and \( y = -2 \). Sketch accuracy may vary, so the exact intersection should show that both lines meet at approximately \((-2, -2)\).

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Key Concepts

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

Graphical Method
When dealing with systems of equations, the graphical method provides a visual approach to finding solutions. This method involves plotting each equation on a graph and seeing where they intersect. By converting equations into lines and sketching them, you can easily observe their intersection point. This technique is particularly useful because:
  • It offers a visual representation of how equations relate.
  • It works well for linear systems but can also be adapted for non-linear ones.
  • It helps in understanding the concept of slope and intercept through visual means.
Observing the intersection graphically also aids in verifying computational solutions obtained by algebraic methods.
Slope-Intercept Form
Slope-intercept form is a common way to express linear equations. In its structure, it's written as: \[ y = mx + b \] Here, \( m \) represents the slope, and \( b \) is the y-intercept of the line. This form makes it easy to plot equations on a graph. With the y-intercept and slope, you can quickly sketch the line. Important highlights:
  • The slope \( m \) indicates how steep the line is. A positive slope means the line inclines, whereas a negative slope indicates declination.
  • The y-intercept \( b \) tells where the line crosses the y-axis. This is your starting point when drawing the graph.
  • This form simplifies graphing and quickly establishing how changes in equations affect the graph.
Implementation of the slope-intercept form is a critical step in graphical solutions, especially when sketching is involved.
Linear Equations
Linear equations represent mathematical expressions resulting in straight lines when graphed. In the context of solving systems, linear equations are pivotal: they express relationships between variables. Each linear equation forms a line based on its coefficients and constants. Key aspects include:
  • They show a constant rate of change. For instance, in \( y = 2x + 2 \), the rate of change of \( y \) with respect to \( x \) is constant, represented by the coefficient of \( x \).
  • Parts of the equation, like the slope and y-intercept, are crucial in determining how the line will be graphed.
  • Graphing these lines elucidates the solution to the system, understood as the intersection point where the graphs meet.
The simplicity of linear equations makes them essential for introducing and understanding more complex mathematical concepts.
Intersection Point
The intersection point is the solution to a system of equations when graphed. It's the coordinates where two lines meet on a graph. This point signifies a common solution to both equations in the system. Understanding it involves:
  • Recognizing that the intersection point satisfies both equations. If you substitute the coordinates into each original equation, both should hold true.
  • Realizing its significance in identifying whether the system has no solution, one solution, or infinitely many solutions.
  • Precision in graphing enhances the accuracy of estimating this point, especially when exact coordinates aren't easily inferred.
In systems of equations involving linear graphs, this intersection point is pivotal in understanding the relationship between the equations.

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Most popular questions from this chapter

Set up appropriate systems of two linear equations in two unknowns and then solve the systems by determinants. All numbers are accurate to at least two significant digits. The velocity of sound in steel is \(15,900 \mathrm{ft} / \mathrm{s}\) faster than the velocity of sound in air. One end of a long steel bar is struck, and an instrument at the other end measures the time it takes for the sound to reach it. The sound in the bar takes 0.0120 s, and the sound in the air takes 0.180 s. What are the velocities of sound in air and in steel?

Set up appropriate systems of two linear equations and solve the systems algebraically. All data are accurate to at least two significant digits. Regarding the forces on a truss, a report stated that force \(F_{1}\) is twice force \(F_{2}\) and that twice the sum of the two forces less 6 times \(F_{2}\) is 6 N. Explain your conclusion about the magnitudes of the forces found from this support.

Set up appropriate systems of two linear equations in two unknowns and then solve the systems by determinants. All numbers are accurate to at least two significant digits. A new development has 3 -bedroom homes and 4-bedroom homes. The developer's profit was \(\$ 25,000\) from each 3 -br home, and \(\$ 35,000\) from each 4 -br home, totaling \(\$ 6,800,000\). Total annual property taxes are \(\$ 560,000,\) with \(\$ 2000\) from each 3 -br home and \(\$ 3000\) from each 4 -br home. How many of each were built?

Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits. A certain amount of a fuel contains 150,000 Btu of potential heat. Part is burned at \(80 \%\) efficiency, and the rest is burned at \(70 \%\) efficiency, such that the total amount of heat actually delivered is 114,000 Btu. Find the amounts burned at each efficiency.

Solve the given systems of equations by determinants. $$\begin{aligned} &3 x-y=3\\\ &4 x=3 y+14 \end{aligned}$$
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