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Chapter 7: Problem 7

Use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(5 x=3 y-15\)

Short Answer

Expert verified
The x and y intercepts of the linear equation \(5 x=3 y-15\) are -3 and 5 respectively. The points (-3,0) and (0,5) are plotted on the graph and a line is drawn through them.

Step by step solution

01

Find the x intercept

To find the x intercept, we'll make y = 0 in the equation \(5x = 3y - 15\). Substituting values, we have \(5x = 3(0) - 15 \Rightarrow 5x = -15 \Rightarrow x = -15 / 5 \Rightarrow x = -3\). So, the x intercept is -3.
02

Find the y intercept

To find the y intercept, make x = 0 in the equation. Hence, \(5(0) = 3y - 15 \Rightarrow 0 = 3y -15 \Rightarrow 15 = 3y \Rightarrow y = 15/3 \Rightarrow y = 5\). So, the y intercept is 5.
03

Graph the equation

Now, plot the x intercept at (-3, 0) and the y intercept at (0, 5) on a graph. Draw a straight line through these points to create a visual representation of the linear equation.

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Key Concepts

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

Understanding the X-Intercept
The x-intercept is where a graph crosses the x-axis. This point happens when the y-value is zero, because you鈥檙e not moving up or down on the graph. To find the x-intercept of a linear equation, you need to set the y-variable to 0 and solve for x.
For the equation given, which is initially in the form of \(5x = 3y - 15\), you'd place zero in place of y:
  • Set \(y = 0\) \(\Rightarrow 5x = 3(0) - 15\)
  • Solve for \(x\): \(5x = -15 \Rightarrow x = \frac{-15}{5} = -3\)
Thus, the x-intercept is the point \((-3, 0)\). This point is crucial because it helps anchor the line on the graph.
Exploring the Y-Intercept
In contrast to the x-intercept, the y-intercept occurs where the graph crosses the y-axis. Here, the x-value is zero, hence you move up or down along the y-axis. To determine the y-intercept, substitute 0 for x in the equation and solve for y.
For the given equation \(5x = 3y - 15\), insert zero for x:
  • Set \(x = 0\) \(\Rightarrow 5(0) = 3y - 15\)
  • Solve for \(y\): \(0 = 3y - 15 \Rightarrow 3y = 15 \Rightarrow y = \frac{15}{3} = 5\)
Therefore, the y-intercept is the point \((0, 5)\). This y-intercept provides another anchor point essential for graphing the line.
Defining a Linear Equation
Linear equations are expressions that represent straight lines when graphed. They typically come in the form \(Ax + By = C\), where A, B, and C are constants.
In the context of this exercise, the equation provided is \(5x = 3y - 15\), which can be rearranged to the standard form \(5x - 3y = -15\). Every linear equation corresponds to a line that has constant rate of change and doesn鈥檛 curve.
Key characteristics include:
  • Consistent rate of change (slope)
  • Straight line
  • Two intercepts: x-intercept and y-intercept
Understanding linear equations helps in predicting values and interpreting data relationships on a graph.
Graphing with Intercepts
Graphing is a visual way to represent equations and explore the relationships between variables. Using the intercepts is a convenient method for graphing linear equations, especially when coordinates are easy to compute.

The process involves the following steps:
  • Identify the x-intercept, which is the point on the graph where the line crosses the x-axis, here at \((-3, 0)\).
  • Identify the y-intercept, which is the point where the line crosses the y-axis, here at \((0, 5)\).
  • Plot these intercepts on a coordinate plane.
  • Draw a straight line through the points.
The line represents all solutions to the linear equation \(5x = 3y - 15\). This technique simplifies the graphing process and helps interpret how changes in x and y affect each other in the equation.

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Most popular questions from this chapter

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \geq 2 \\ y \leq 3\end{array}\right.\)

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|r|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & 4 \\ \hline 1 & 5 \\ \hline 2 & 7 \\ \hline 3 & 11 \\ \hline 4 & 19 \\ \hline \end{array} $$

A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, \(y\), in feet, in terms of its horizontal distance, \(x\), in feet. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { x, Ball's } \\ \text { Horizontal } \\ \text { Distance } \\ \text { (feet) } \end{array} & \begin{array}{c} \boldsymbol{y} \text {, Ball's } \\ \text { Height } \\ \text { (feet) } \end{array} \\ \hline 0 & 6 \\ \hline 0.5 & 7.4 \\ \hline 1.5 & 9 \\ \hline 4 & 6 \\ \hline \end{array} $$ QuadReg \(y=a x^{2}+b x+c\) \(a=-.8\) \(\mathrm{b}=3.2\) \(c=6\)

In your own words, describe how to solve a linear programming problem.

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is \(\$ 125\) for the rearprojection televisions and \(\$ 200\) for the plasma televisions. a. Let \(x=\) the number of rear-projection televisions manufactured in a month and \(y=\) the number of plasma televisions manufactured in a month. Write the objective function that describes the total monthly profit. b. The manufacturer is bound by the following constraints: \- Equipment in the factory allows for making at most 450 rear-projection televisions in one month. \- Equipment in the factory allows for making at most 200 plasma televisions in one month. \- The cost to the manufacturer per unit is \(\$ 600\) for the rear-projection televisions and \(\$ 900\) for the plasma televisions. Total monthly costs cannot exceed \(\$ 360,000 .\) Write a system of three inequalities that describes these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) must both be nonnegative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200),(300,200)\), \((450,100)\), and \((450,0)\).] e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing rear-projection televisions each month and plasma televisions each month. The maximum monthly profit is $\$$
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