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Chapter 4: Problem 88

In the following exercises, graph by plotting points. \(5 x+y=6\)

Short Answer

Expert verified
Plot points (0, 6), (1, 1), (2, -4), (-1, 11), and (-2, 16); then draw the line through them.

Step by step solution

01

Rewrite the equation in slope-intercept form

The given equation is: \[5x + y = 6\] Isolate y: \[y = -5x + 6\]
02

Choose values for x

Choose different values for x to find corresponding values of y. Common choices are: x = 0, 1, 2, -1, -2
03

Calculate corresponding y values

Using the equation \[y = -5x + 6\]: For x = 0: \[y = -5(0) + 6 = 6\] For x = 1: \[y = -5(1) + 6 = 1\] For x = 2: \[y = -5(2) + 6 = -4\] For x = -1: \[y = -5(-1) + 6 = 11\] For x = -2: \[y = -5(-2) + 6 = 16\]
04

Plot the points on graph

Plot the points calculated in Step 3 on a coordinate plane: (0, 6), (1, 1), (2, -4), (-1, 11), (-2, 16)
05

Draw the line

Connect the points with a straight line. This is the graph of the equation \[5x + y = 6\]

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Key Concepts

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

slope-intercept form
The slope-intercept form of a linear equation makes it easy to graph the equation. The form is written as \[y = mx + b\] where \( m \) is the slope, and \( b \) is the y-intercept. The slope \( m \) indicates how steep the line is and the direction it goes. The y-intercept \( b \) is where the line crosses the y-axis. For example, when converting the equation \[ 5x + y = 6 \] to slope-intercept form, you isolate \(y\) to get \[ y = -5x + 6\]. Here, \( m = -5 \) and \( b = 6 \). This tells you the line slopes downward 5 units for every unit it moves to the right, and it crosses the y-axis at 6.
plotting points
Plotting points helps us visualize solutions to equations. To plot points from a linear equation:
  • Select values for \( x \)
  • Calculate corresponding \( y \) values
  • Mark the points on the coordinate plane
    • For the equation \( y = -5x + 6 \), we can choose \( x = 0, 1, 2, -1, -2 \). Find \( y \) for each value:
    • When \(x = 0\), \( y = 6\)
    • When \(x = 1\), \( y = 1\)
    • When \(x = 2\), \( y = -4\)
    • When \(x = -1\), \( y = 11\)
    • When \(x = -2\), \( y = 16\)
    Plot these points: (0, 6), (1, 1), (2, -4), (-1, 11), (-2, 16).
coordinate plane
A coordinate plane is the surface on which we graph points, lines, and curves. It is made up of two axes:
  • The horizontal axis (x-axis)
  • The vertical axis (y-axis)
Where these axes intersect is called the origin (0,0). Each point in the plane can be described by an ordered pair \((x, y)\). Using our points from the equation \( y = -5x + 6 \):
  • (0, 6)
  • (1, 1)
  • (2, -4)
  • (-1, 11)
  • (-2, 16)
You plot these points by starting at the origin, moving along the x-axis, and then moving up or down to the y-coordinate.
linear equations
Linear equations represent straight lines. They involve variables raised to the power of one and can usually be written in forms such as \[y = mx + b\] or \[Ax + By = C\]. The equation \( 5x + y = 6 \) can be rewritten in slope-intercept form \[ y = -5x + 6 \], showing it is a linear equation. Linear equations are predictable and have a constant rate of change, as seen in the slope. When plotted on a graph, they form a straight line. By connecting points such as (0, 6), (1, 1), and other calculated points, you will see the linearity of the relationship.

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