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Chapter 3: Problem 68

(a) find three solutions of the equation. (b) graph the equation. \(y=-x+7\)

Short Answer

Expert verified
Three solutions are (0, 7), (2, 5), (4, 3). Graph is a straight line going downwards from left to right.

Step by step solution

01

- Select values for x

Choose three different values for x. For this example, let's use x = 0, x = 2, and x = 4.
02

- Plug in the values of x to find corresponding y

Substitute x into the equation to find the corresponding y values for each selected x.When x = 0:\[ y = -0 + 7 = 7 \]When x = 2:\[ y = -2 + 7 = 5 \]When x = 4:\[ y = -4 + 7 = 3 \]
03

- Write down the solutions

List the x and y pairs (solutions) found in Step 2. The solutions are:1. (0, 7)2. (2, 5)3. (4, 3)
04

- Draw the coordinate axes

Plot the x-axis and y-axis on graph paper or graphing software.
05

- Plot the points

Plot the points found in Step 3 on the coordinate plane:1. (0, 7)2. (2, 5)3. (4, 3)
06

- Draw the line

Connect the plotted points with a straight line. This line represents the equation \( y = -x + 7 \).

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Key Concepts

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

coordinate plane
The coordinate plane is an essential concept in graphing linear equations, simplifying the visualization of algebraic expressions. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis), intersecting at the origin (0,0). Points are plotted using ordered pairs (x, y), where x indicates the horizontal position and y the vertical position.
The plane is divided into four quadrants, each with a distinct sign for x and y values:
  • Quadrant I: (positive x, positive y)
  • Quadrant II: (negative x, positive y)
  • Quadrant III: (negative x, negative y)
  • Quadrant IV: (positive x, negative y)
Understanding the coordinate plane is crucial for graphing any equation, as it helps determine the position of points and the shape of their connection.
graphing linear equations
Graphing linear equations translates algebraic expressions into visual representations. Here's a step-by-step process:
Select values for x. Any values work, but simple ones like 0, 1, and 2 make calculations easier.
Substitute x into the equation. Using the equation provided, calculate the corresponding y values. For example, if x = 0, then \( y = -0 + 7 = 7 \). List all ordered pairs (x, y).
Plot the points on the coordinate plane. Mark each pair with a dot. So, plot (0, 7), (2, 5), and (4, 3).
Connect the points with a straight line. This line will demonstrate the relationship between x and y as described by the equation. Linear equations always produce straight lines.
Properly graphing reveals patterns and intersections that clarify how variables relate.
solving for y
Solving for y in a linear equation is a key step in plotting points. Given an equation in the form \( y = mx + b \), follow these steps:
Identify the equation's structure: Look for the slope (m) and y-intercept (b). For \( y = -x + 7 \), m = -1, and b = 7.
Select x values. Typically, choose simple numbers like 0, \pm1\ or \pm2\.
Substitute x values and solve. For example:
  • If x = 0: \( y = -0 + 7 = 7 \)
  • If x = 2: \( y = -2 + 7 = 5 \)
  • If x = 4: \( y = -4 + 7 = 3 \)
Write down the solutions: These will be your coordinates to plot. Understanding \( y = mx + b \) helps graph any line, making complex algebra simpler and more tangible.

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Most popular questions from this chapter

(a) write the equation of the horizontal line that passes through the point. (b) graph the equation. \((-2,-1)\)

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

The New York City Feedback Citywide Customer Survey Report was published in 2008 . The response rate describes the percent of mailed questionnaires that were completed and returned. Find the number of mailed questionnaires. Round to the nearest whole number. \- 24,339 questionnaires were completed and returned \- \(18 \%\) response rate (Source: www.nyc.gov)

For exercises 95-98, 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 \(y\)-intercept of \(6 x+5 y=30\). Incorrect Answer: \(9 x+2 y=36\) $$ \begin{aligned} 9(0)+2 y &=36 \\ 2 y &=36 \\ \frac{2 y}{2} &=\frac{36}{2} \\ y &=18 \end{aligned} $$ The \(x\)-intercept is \((0,18)\).

(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. \((-1,-3) ;(-4,-1)\)
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