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Chapter 3: Problem 70

(a) find three solutions of the equation. (b) graph the equation. \(y=10 x+50\)

Short Answer

Expert verified
The solutions are (-5, 0), (0, 50), and (5, 100). The graph is a straight line with a positive slope.

Step by step solution

01

- Choose values for x

Select three different values for the variable x. For simplicity, let's choose x = -5, x = 0, and x = 5.
02

- Calculate y for each x value

Substitute each chosen value of x into the equation and solve for y.1. For x = -5:\[ y = 10(-5) + 50 = -50 + 50 = 0 \]2. For x = 0:\[ y = 10(0) + 50 = 0 + 50 = 50 \]3. For x = 5:\[ y = 10(5) + 50 = 50 + 50 = 100 \]
03

- List the solutions

The solutions to the equation are the ordered pairs (x, y) corresponding to the chosen x values and their calculated y values. These are (-5, 0), (0, 50), and (5, 100).
04

- Create a graph of the equation

Plot the points (-5, 0), (0, 50), and (5, 100) on a coordinate plane. Draw a straight line through these points to represent the equation. The graph will show a straight line with a positive slope starting from (0, 50) and increasing as x increases.

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Key Concepts

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

solving equations
Solving equations is a fundamental skill in algebra. The goal is to find values for the variable(s) that make the equation true. In the given exercise, we start with the equation of a line,
\(y = 10x + 50\).
To find solutions:
  • Select values for x.
  • Substitute these x values into the equation.
  • Solve for the corresponding y values.

To illustrate, if we choose \(x = -5\), substituting it into the equation gives us:
y = 10(-5) + 50 = 0.
Therefore, one solution is (-5, 0). < br> By using different x values such as 0 and 5, we can determine y values and thus find more solutions. This method helps us understand the relationship between x and y in the equation.

Simple and systematic substitutions can make solving equations straightforward and logical.
graphing linear equations
Graphing linear equations visually represents the relationship between variables. In our example, \(y = 10x + 50\), each solution pair \((x, y)\) plots a point on the graph.

Here is the step-by-step process for graphing:
  • Find three or more solution points by substituting different x values into the equation.
  • For instance, points (-5, 0), (0, 50), and (5, 100) are solutions.
  • Plot these points on the coordinate plane.
  • Draw a straight line through these plotted points, which represents the equation.
< br>The straight line indicates that the variables y and x have a linear relationship. The slope of the line (\(m = 10\)) shows that y increases by 10 units for each unit increase in x.

Graphing provides a visual understanding of how the values of x and y interact within the equation.
coordinate plane
The coordinate plane is a two-dimensional surface where we plot points, lines, and curves. It consists of two perpendicular axes:
  • The horizontal axis (x-axis).
  • The vertical axis (y-axis).
The point where these axes intersect is known as the origin (0, 0). Each point on the plane is defined by an ordered pair (x, y). For example, in the equation \(y = 10x + 50\), solutions like (0, 50) represent points on the coordinate plane.

To plot points:
  • Start from the origin.
  • Move x units along the x-axis.
  • Then move y units along the y-axis.
  • Mark the point.


This system helps visualize mathematical relationships and shapes. Understanding how to navigate a coordinate plane builds a foundation for graphing equations and analyzing data.

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Most popular questions from this chapter

(a) find three solutions of the equation. (b) graph the equation. \(y=-\frac{1}{4} x+3\)

Use the slope formula to find the slope of the line that passes through the points. \((-5,4) ;(-9,3)\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(2 x+5 y=15\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((1,5) ;(3,-3)\)

For exercises 89-92, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Find the slope of the line that passes through \((7,1)\) and \((9,4)\). Incorrect Answer: \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) $$ \begin{aligned} m &=\frac{9-7}{4-1} \\ m &=\frac{2}{3} \end{aligned} $$
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