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Chapter 5: Problem 27

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. $$\begin{aligned} &-2 r_{1}+2 r_{2}=7\\\ &4 r_{1}-2 r_{2}=1 \end{aligned}$$

Short Answer

Expert verified
The solution to the system is approximately \((4, 7.5)\).

Step by step solution

01

Convert Equations to Slope-Intercept Form

First, we need to convert each equation in the system to the slope-intercept form, which is \( y = mx + b \). For the first equation \(-2r_1 + 2r_2 = 7\), solve for \( r_2 \) to get \( r_2 = r_1 + 3.5 \). For the second equation \(4r_1 - 2r_2 = 1\), solve for \( r_2 \) to get \( r_2 = 2r_1 - 0.5 \).
02

Identify Slopes and Intercepts

In our slope-intercept equations, \( r_2 = r_1 + 3.5 \) has a slope \( m = 1 \) and a y-intercept \( b = 3.5 \). The equation \( r_2 = 2r_1 - 0.5 \) has a slope \( m = 2 \) and a y-intercept \( b = -0.5 \). These characteristics will help us sketch the lines on the graph.
03

Sketch the Graphs

On a coordinate plane, plot the y-intercepts \((0, 3.5)\) and \((0, -0.5)\) for each line. Use the slopes to determine another point on each line. For \( r_2 = r_1 + 3.5 \), move up 1 and right 1 from \( (0, 3.5) \). For \( r_2 = 2r_1 - 0.5 \), move up 2 and right 1 from \( (0, -0.5) \). Sketch the lines extending through these points.
04

Determine the Intersection Point

The solution of the system is the point where these two lines intersect. By comparing their equations and the graph, find that the lines intersect at \((4, 7.5)\). The point \( (4, 7.5) \) is the estimated solution to the nearest 0.1.

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Key Concepts

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

Graphical Method
The graphical method to solve systems of equations involves drawing each equation as a line on a graph. The point where the lines intersect is the solution to the system. Unlike algebraic methods that manipulate equations to find a solution, the graphical method provides a visual representation.
  • Each equation is simplified into a form that makes it easy to draw on a coordinate plane, typically the slope-intercept form.
  • Lines are plotted using critical points like the y-intercept and additional points determined by the slope.
  • By observing where the lines cross, the intersection point is identified as the solution.
This method is particularly useful in visualizing how solutions to equations relate to each other. It helps you understand the concept of solutions lying at the intersection, making complex algebraic ideas more concrete.
Slope-Intercept Form
The slope-intercept form of a line is a simple yet powerful way to express linear equations. It shows the relationship between the slope of the line and its y-intercept through the equation: \( y = mx + b \).
Where:
  • \( m \) represents the slope of the line, indicating its steepness and direction. A positive slope rises, while a negative slope falls.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis. It's the starting value of \( y \) when \( x = 0 \).
Converting equations into this form assists in easily identifying characteristics necessary for graphing. It tells us exactly how to draw the line: where it begins, and how it progresses across the graph. This is vital in the graphical method, as it simplifies the work needed to plot each line on the coordinate plane.
Line Intersection
In the context of solving systems of equations, the point of line intersection is critical. This point marks the location where the equations, when graphed, meet. It's the shared solution for both equations.
  • The intersection point gives the common values for variables that satisfy both equations.
  • If two lines intersect at a point, we have a unique solution to the system.
  • If lines are parallel, there is no intersection, indicating no common solution.
Graphing allows us to precisely determine where the lines cross. It's crucial to accurately estimate or calculate this intersection point to ensure the solution is correct. Observing intersection points visually reinforces the concept of equivalence across equations in a system.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we graphically represent equations, specifically linear ones for this exercise.
  • It consists of a horizontal axis (x-axis) and a vertical axis (y-axis), intersecting at a point called the origin (0,0).
  • Any point on this plane is represented by a pair of numbers, known as coordinates: \( (x, y) \).
  • It's essential for plotting the solutions and behaviors of equations, providing a visual framework to observe relationships and intersections.
Understanding the coordinate plane is foundational to graphing. It enables us to accurately draw points, lines, and curves, which are crucial in demonstrating how equations interact through the graphical method.

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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. since ancient times, a rectangle for which the length \(L\) is approximately 1.62 times the width \(w\) has been considered the most pleasing to view, and is called a golden rectangle. If a painting in the shape of a golden rectangle has a perimeter of \(4.20 \mathrm{m}\), find the dimensions.

Set up appropriate systems of two linear equations and solve the systems algebraically. All data are accurate to at least two significant digits. In an election, candidate \(A\) defeated candidate \(B\) by 2000 votes. If 1.0 \(\$ 6\) of those who voted for \(A\) had voted for \(B\), \(B\) would have won by 1000 votes. How many votes did each receive?

Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits. A manufacturer produces three models of DVD players in a year. Four times as many of model A are produced as model C. and 7000 more of model \(B\) than model \(C\). If the total production for the year is 97,000 units, how many of each are produced?

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. Two types of electromechanical carburetors are being assembled and tested. Each of the first type requires 15 min of assembly time and 2 min of testing time. Each of the second type requires \(12 \mathrm{min}\) of assembly time and 3 min of testing time. If 222 min of assembly time and 45 min of testing time are available, how many of each type can be assembled and tested, if all the time is used?

Evaluate the given determinants. $$\left|\begin{array}{cc} -8 & -4 \\ -32 & 16 \end{array}\right|$$
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