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Chapter 10: Problem 51

Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for \(y\) in terms of \(x\) before graphing if you are using a graphing calculator.) Solve the system either by zooming in and using \([\text { TRACE }]\) or by using Int er sect. Round your answers to two decimals. $$\left\\{\begin{array}{l} 0.21 x+3.17 y=9.51 \\ 2.35 x-1.17 y=5.89 \end{array}\right.$$

Short Answer

Expert verified
Intersection at approximately (2.44, 2.84).

Step by step solution

01

Rewrite Each Equation in Slope-Intercept Form

We start by solving each equation for \( y \). For the first equation: \( 0.21x + 3.17y = 9.51 \). Subtract \( 0.21x \) from both sides to get \( 3.17y = -0.21x + 9.51 \). Divide everything by \( 3.17 \) to isolate \( y \): \( y = -\frac{0.21}{3.17}x + \frac{9.51}{3.17} \). Similarly, the second equation: \( 2.35x - 1.17y = 5.89 \). Subtract \( 2.35x \) from both sides: \( -1.17y = -2.35x + 5.89 \). Divide everything by \(-1.17\): \( y = \frac{2.35}{1.17}x - \frac{5.89}{1.17} \).
02

Calculate the Slopes and Intercepts

For the first equation, \( y = -\frac{0.21}{3.17}x + \frac{9.51}{3.17} \), calculate the slope \( m_1 = -0.06624 \) and the y-intercept \( b_1 = 3.00 \). For the second equation, \( y = \frac{2.35}{1.17}x - \frac{5.89}{1.17} \), the slope \( m_2 = 2.009 \) and the y-intercept \( b_2 = -5.03 \).
03

Graph Both Equations

Using a graphing calculator or software, input the equations: \( y = -0.06624x + 3.00 \) and \( y = 2.009x - 5.03 \). Ensure that both lines are drawn in the same viewing window to clearly see their intersection.
04

Identify the Intersection Point

Using the graphing device, either employ the 'Trace' feature to navigate to the intersection point or use the 'Intersect' function to calculate the exact point where the lines intersect. This point will give the solution to the system of equations.
05

Round and Report the Intersection Coordinates

The calculated intersection from the graphing calculator gives the exact point \( (x, y) \). Ensure the intersection coordinates are rounded to two decimal places for each value. For example, if the calculator gives you (2.654, 1.234), this should be rounded to (2.65, 1.23).

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Key Concepts

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

Understanding Slope-Intercept Form
The slope-intercept form is an essential concept in graphing linear equations. This form helps express a linear equation in a way that is easy to visualize and graph. It is generally written as: \[ y = mx + b \]. Here:
  • \(m\) represents the slope of the line. This tells us how steep the line is and the direction in which it moves. A positive slope means the line goes upwards, whereas a negative slope means it goes downwards as you move from left to right.
  • \(b\) is the y-intercept, which is where the line crosses the y-axis. This gives you a starting point on the graph.
To write a linear equation in slope-intercept form, you need to solve for \(y\). In this exercise, both equations were converted into \(y = mx + b\) form, making it simpler to identify their slopes and intercepts. By doing so, you can graph them directly using a calculator or graphing tool.
Exploring a System of Equations
A system of equations consists of two or more equations with the same set of variables. In this scenario, we have two linear equations involving the variables \(x\) and \(y\). The goal is to find the values of \(x\) and \(y\) which satisfy both equations at the same time. The system can be visualized as two lines when graphed. The solution to the system is typically the intersection point of these two lines. It represents the \((x, y)\) coordinates that fit both equations simultaneously. It鈥檚 critical because:
  • If the lines intersect at a single point, the system has one solution.
  • If the lines are parallel, there is no solution.
  • If the lines coincide, there are infinitely many solutions.
Graphing provides a visual method to see where the solutions may lie, making it easier to comprehend the relationships between the equations in the system.
Finding the Intersection Point
The intersection point is critical in solving a system of linear equations graphically. It is where the graphs of the equations meet in the coordinate plane. In simpler terms, it is the solution to both equations. To find this point graphically, you can:
  • Use the TRACE function on a graphing calculator. This lets you manually steer along one of the graph lines until you approximate the intersection point.
  • Use the INTERSECT feature if the calculator has it, which provides a more precise calculation of where the two lines touch.
Once the intersection is identified, its coordinates should be rounded as needed. This step ensures clarity and precision in describing the result of the system, as in needing to round numbers to two decimal places, like (2.65, 1.23).
Utilizing a Graphing Calculator
Graphing calculators are powerful tools for visualizing mathematical relationships and solving equations. They simplify the process of graphing lines and identifying intersections without extensive manual calculations. Here is a rundown of how they work in the context of the exercise:
  • They allow you to input multiple equations, such as those rewritten in slope-intercept form.
  • They display the equations' graphs on a coordinate plane, facilitating a visual comparison.
  • Using features like TRACE or INTERSECT, they can determine the intersection point efficiently.
For students, these calculators are invaluable not only in plotting graphs but also in ensuring that solutions to equations are precise and conducted with ease. They are an excellent resource for both verifying work and exploring additional mathematical problems.

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Most popular questions from this chapter

Boat Speed \(A\) boat on a river travels downstream between two points, \(20 \mathrm{mi}\) apart, in \(1 \mathrm{h}\). The return trip against the current takes \(2 \frac{1}{2} \mathrm{h} .\) What is the boat's speed, and how fast does the current in the river flow?

Find the partial fraction decomposition of the rational function. $$\frac{2 x+1}{x^{2}+x-2}$$

Investments A man invests his savings in two accounts, one paying \(6 \%\) and the other paying \(10 \%\) simple interest per year. He puts twice as much in the lower-yielding account because it is less risky. His annual interest is \(\$ 3520 .\) How much did he invest at each rate?

Write the system of equations as a matrix equation (see Example 6). $$\left\\{\begin{aligned} x-y+z &=2 \\ 4 x-2 y-z &=2 \\ x+y+5 z &=2 \\\\-x-y-z &=2 \end{aligned}\right.$$

Use Cramer's Rule to solve the system. $$\left\\{\begin{aligned} \frac{1}{3} x-\frac{1}{5} y+\frac{1}{2} z &=\frac{7}{10} \\ -\frac{2}{3} x+\frac{2}{5} y+\frac{3}{2} z &=\frac{11}{10} \\ x-\frac{4}{5} y+z &=\frac{9}{5} \end{aligned}\right.$$
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