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Chapter 3: Problem 35

Use the intercept method to graph each equation. $$ 30 x+y=-30 $$

Short Answer

Expert verified
The x-intercept is \((-1,0)\) and the y-intercept is \((0,-30)\). Graph these points and draw the line through them.

Step by step solution

01

Identify the Equation Format

The given equation is \(30x + y = -30\). This is a linear equation in standard form \(Ax + By = C\), where \(A = 30\), \(B = 1\), and \(C = -30\).
02

Find the X-Intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\): \(30x + 0 = -30\). This simplifies to \(30x = -30\). Dividing both sides by 30 gives \(x = -1\). The x-intercept is \((-1, 0)\).
03

Find the Y-Intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\): \(30(0) + y = -30\). This simplifies to \(y = -30\), so the y-intercept is \((0, -30)\).
04

Graph the Intercepts

Plot the x-intercept \((-1, 0)\) and the y-intercept \((0, -30)\) on a coordinate plane. These two points will determine the line.
05

Draw the Line

Draw a straight line through the plotted intercepts. This line represents the equation \(30x + y = -30\). Ensure it extends across the entire graph, showing the infinite nature of the line.

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Key Concepts

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

Linear Equations
A linear equation is a type of equation that forms a straight line when graphed on a coordinate plane. It can usually be written in the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. In this format, \(x\) and \(y\) are variables that represent points on a graph.
Linear equations have many properties:
  • They create straight lines when plotted.
  • Their solutions are the sets of all points \( (x, y) \) that lie on the line.
  • They can have infinitely many solutions, as the line extends indefinitely in both directions.
  • You can graph them using points like intercepts, which are locations where the line crosses the axes.
Understanding how to write and manipulate these equations is important for solving mathematical problems related to lines. They can describe many real-world situations where a constant rate of change is involved, such as distance over time or cost per item.
X-Intercept
The x-intercept is a key feature of any graphed line. It refers to the point at which the line crosses the x-axis on a graph.
To find this intercept, you simply set the \(y\) variable to zero in the equation and solve for \(x\):
  • This method effectively "removes" y from the equation because, at the x-intercept, \(y = 0\).
  • For instance, with the equation \(30x + y = -30\), setting \(y = 0\) leads us to \(30x = -30\), so \(x = -1\). Thus, the x-intercept is \((-1, 0)\).
Finding the x-intercept is straightforward and offers a helpful anchor point which, along with the y-intercept, can help quickly establish the overall slope and position of the line on a graph.
Y-Intercept
The y-intercept is another foundational aspect when working with linear equations. It is the point where the line crosses the y-axis, and happens when \(x = 0\).
To locate the y-intercept, you substitute \(x = 0\) into the equation:
  • By making \(x\) zero, you focus on what happens purely along the y-axis.
  • Using the example equation \(30x + y = -30\), substituting \(x = 0\) gives \(y = -30\), thus the y-intercept is \((0, -30)\).
This point is critical because, together with the x-intercept, it helps in easily plotting the linear equation on a coordinate plane. It also represents the starting point in many real-world applications, like the initial amount of a resource before time or a process begins.
Graphing
Graphing is the process of representing algebraic equations visually on a coordinate plane. This step is particularly beneficial for understanding the relationship and interaction between variables.
To graph a linear equation using intercepts:
  • First, you find the x-intercept by setting \(y = 0\).
  • Then, determine the y-intercept by setting \(x = 0\).
  • Once both intercepts, \((-1, 0)\) and \((0, -30)\) for our example, are found, you plot them on the graph.
  • With these two plotted points, you draw a straight line through them.
The line produced should stretch across the graph, demonstrating the infinite nature of a linear equation. Graphing in this way provides a clear visualization of how solutions to the equation relate to each other and helps solve and understand more complex mathematical problems or scenarios.

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Most popular questions from this chapter

Mrs. Cansino has a choice of two babysitters. Sitter 1 charges \(\$ 6\) per hour, and Sitter 2 charges S7 per hour. If \(x\) represents the number of hours she uses Sitter 1 and \(y\) represents the number of hours she uses Sitter \(2,\) the graph of \(6 x+7 y \leq 42\) shows the possible ways she can hire the sitters and not spend more than \(\$ 42\) per week. Graph the inequality. Then find three possible ways she can hire the babysitters so that her weekly budget for babysitting is not exceeded.

Medications. A doctor prescribes an ointment that is \(2 \%\) hydrocortisone. A pharmacist has \(1 \%\) and \(5 \%\) concentrations in stock. How many ounces of each should the pharmacist use to make a 1 -ounce tube?

In the function \(y=-5 x+2,\) why do you think the value of \(x\) is called the input and the corresponding value of \(y\) the output?

Parts Lists. The function \(f(r)=2.30+3.25(r+0.40)\) approximates the length (in feet) of the belt that joins the two pulleys, where \(r\) is the radius (in feet) of the smaller pulley. Find the belt length needed for each pulley in the parts list. CAN'T COPY THE IMAGE $$ \begin{array}{|c|c|} \hline & {\text { Parts list }} \\ \hline \text { Pulley } & {r} & {\text { Belt length }} \\ \hline P-45 M & {0.32} \\ {P-08 D} & {0.24} \\ \hline \end{array} $$

$$ \text { Simplify: } 40\left(\frac{3}{8} x-\frac{1}{4}\right)+40\left(\frac{4}{5}\right) $$
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