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Chapter 11: Problem 49

\(49-52\) 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 rounded to two decimal places, either by zooming in and using [TRACE] or by using Intersect. $$ \left\\{\begin{array}{l}{0.21 x+3.17 y=9.51} \\ {2.35 x-1.17 y=5.89}\end{array}\right. $$

Short Answer

Expert verified
The solution to the system is approximately \((4.01, 2.73)\).

Step by step solution

01

Solve for y in the First Equation

To rearrange the equation \(0.21x + 3.17y = 9.51\) for \(y\), we start by isolating \(y\). Move \(0.21x\) to the other side: \(3.17y = -0.21x + 9.51\). Then, divide the entire equation by \(3.17\): \(y = -\frac{0.21}{3.17}x + \frac{9.51}{3.17}\). Simplify to get \(y = -0.0663x + 3.00\).
02

Solve for y in the Second Equation

Take the second equation \(2.35x - 1.17y = 5.89\) and solve for \(y\). Move \(2.35x\) to the other side: \(-1.17y = -2.35x + 5.89\). Divide by \(-1.17\): \(y = \frac{2.35}{1.17}x - \frac{5.89}{1.17}\). Simplify to get \(y = 2.01x - 5.03\).
03

Graph the Equations

Using a graphing device, plot the two equations in the same viewing window. The first equation is \(y = -0.0663x + 3.00\) and the second is \(y = 2.01x - 5.03\). Set an appropriate range for both axes to ensure both lines are clearly visible.
04

Find the Intersection Point

Use the graphing tool's Intersect feature to find where the two lines intersect. Alternatively, zoom in on the graph near the intersection point and use the [TRACE] function to get an accurate value. The lines intersect at approximately \((4.01, 2.73)\).
05

Round the Coordinates

The intersection point based on the graph is \((4.01, 2.73)\). These coordinates are already rounded to two decimal places, as specified by the exercise.

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Key Concepts

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

Graphing Linear Equations
Graphs are visual representations of equations that help us understand the relationships between variables involved. Linear equations form straight lines when graphed and are typically written in the slope-intercept form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. To graph a linear equation, follow these steps:
  • Solve the equation for \( y \), if necessary, to get it in slope-intercept form.
  • Identify the y-intercept \( b \), and plot it on the y-axis.
  • Use the slope \( m \) to find another point. Remember, slope is "rise over run," indicating how much the line rises vertically for a given horizontal movement.
  • Draw a straight line through the points.
These steps will give you a clear visual of the linear equation, from which you can analyze further, such as finding where it intersects another line.
Intersection of Lines
When two lines intersect on a graph, it means they have a meeting point in common, where the equations are both satisfied. This point is called the intersection point and is often denoted as \((x, y)\). The intersection point is the solution to the system of equations represented by the two lines.
  • The solution point satisfies both equations; therefore, when you substitute \( x \) and \( y \) from this point into the original equations, both should hold true.
  • In real-life scenarios, finding the intersection can represent solving problems where two conditions need to be met simultaneously.
  • Graphically locating the intersection may involve accurately drawing both lines and finding the coordinates where they meet.
If intersecting lines are parallel, they do not meet, indicating no solution. If they coincide, it means there are infinitely many solutions.
Using a Graphing Calculator
Graphing calculators are powerful tools that can graph equations and solve systems visually. They can handle the plotting of curves and calculation of intersection points efficiently. Here鈥檚 a straightforward method of using a graphing calculator for solving systems of equations:
  • First, convert the equations to \( y = mx + b \) form, as graphing calculators typically require this format.
  • Enter each equation into the calculator鈥檚 graphing function.
  • Set a suitable window size to ensure both lines are within view.
  • Use the calculator's "INTERSECT" function to pinpoint the exact intersection point. This function computes the solution of the system by finding the \((x, y)\) coordinate where both equations meet.
  • Alternatively, use the "TRACE" function to manually navigate to the intersection point for reading coordinates visually.
Practicing these steps not only helps with solving system equations but enhances understanding of graphing technology.
Isolation of Variables
The isolation of a variable is a key algebraic technique used to rearrange equations so that one variable is alone on one side of the equation. This is a critical step before graphing equations or solving for unknowns. Here鈥檚 how isolation of \( y \) is generally approached:
  • To isolate \( y \), first move all terms not containing \( y \) to the opposite side of the equation. This usually involves adding or subtracting terms.
  • After isolating terms with the variable, divide every part of the equation by the coefficient of \( y \) to solve for \( y \).
  • Rearrange the equation to reveal \( y \) in terms of other constants and variables, giving the slope-intercept form \( y = mx + b \).
  • This preparation allows for straightforward graphing or inputting into calculators.
By mastering this technique, students can handle more complex algebraic tasks and facilitate easier problem-solving and graph interpretations.

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Most popular questions from this chapter

Solve for \(x\) $$ \left|\begin{array}{lll}{1} & {0} & {x} \\ {x^{2}} & {1} & {0} \\ {x} & {0} & {1}\end{array}\right|=0 $$

Use Cramer鈥檚 Rule to solve the system. $$ \left\\{\begin{array}{l}{10 x-17 y=21} \\ {20 x-31 y=39}\end{array}\right. $$

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don鈥檛 calculate the inverse. $$ \left[\begin{array}{rrr}{1} & {2} & {5} \\ {-2} & {-3} & {2} \\ {3} & {5} & {3}\end{array}\right] $$

\(47-50\) . Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place. $$ \left\\{\begin{array}{l}{y \geq x^{3}} \\ {2 x+y \geq 0} \\ {y \leq 2 x+6}\end{array}\right. $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} x^{2}+y^{2} & \leq 8 \\ x & \geq 2 \\ y & \geq 0 \end{aligned}\right. $$
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