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Chapter 9: Problem 17

In Exercises \(7-24,\) find the general form of the equation of the line satisfying the conditions given and graph the line. With \(x\) -intercept \((-5,0)\) and slope 2

Short Answer

Expert verified
y = 2x + 10.

Step by step solution

01

Identify key information

The exercise provides the x-intercept of the line as (-5, 0) and the slope as 2.
02

Recall the point-slope form equation

The point-slope form of a line's equation is given by: y - y_1 = m(x - x_1), where m is the slope and (x_1, y_1) is a point on the line.
03

Substitute the given values

Using the point-slope form equation y - y_1 = m(x - x_1) with: m =2 and (x_1, y_1) = (-5, 0): y - 0 =2(x - (-5)).
04

Simplify the equation

Simplify the equation: y = 2(x + 5).
05

Convert to slope-intercept form

Distribute to obtain the equation in the slope-intercept form (y = mx + b): y = 2x + 10.
06

Graph the line

To graph the line, start at the x-intercept (-5, 0). Use the slope of 2 to find another point on the line: Move up 2 units and right 1 unit from (-5, 0) to get (-4, 2). Draw the line through these two points.

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Key Concepts

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

x-intercept
The x-intercept is where a line crosses the x-axis. At this point, the value of y is always 0. For our example, the x-intercept is given as (-5, 0). This means when the line intersects the x-axis, the x-coordinate is -5.

To find the equation of the line with a known x-intercept and slope, you can use this point as one of the key components. Understanding the x-intercept is crucial because it helps to easily plot the graph and to convert between different forms of the linear equation.
point-slope form
The point-slope form of a line鈥檚 equation is particularly useful when you know one point on the line and the slope. It is written as: \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) is a known point and m is the slope.

In our exercise: \((x_1, y_1) = (-5, 0)\) and m = 2.

Substituting these into the point-slope formula gives us: \( y - 0 = 2(x - (-5)) \), which simplifies to \( y = 2(x + 5) \).

This form is useful for quickly writing the equation of a line when you don't have the y-intercept readily available.
slope-intercept form
The slope-intercept form of a linear equation is one of the most common ways to express the equation of a line. It has the form: \( y = mx + b \), where m is the slope and b is the y-intercept.

From our simplified point-slope equation \( y = 2(x + 5) \), distributing 2 over \( (x+5) \) results in: \( y = 2x + 10 \).

Here, the slope m = 2 and the y-intercept b = 10. This form is great for quickly identifying the steepness and vertical position of a line.
graphing lines
Graphing a line requires understanding both the slope and intercepts of the line.
  • Start by plotting the x-intercept, given in our case as (-5, 0).
  • Then, using the slope, move vertically and horizontally to mark another point. With slope m = 2, for every 1 unit you move horizontally, you move 2 units vertically.
  • From (-5, 0), moving up 2 units and right 1 unit lands us on the point (-4, 2).
Drawing a line through these points will give the graph of the linear equation. This visual method helps solidify the concept of slope and intercept and how they influence the line's position and angle on the graph.

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Most popular questions from this chapter

The distance between \((-3,2)\) and \((d, 2)\) is \(6 .\) Find all the possible values of \(d .\)

Find the midpoint of the line segment between the points given. \((6,11)\) and \((5,-3)\)

Plot the points corresponding to the ordered pairs \(A(2,5), B(3,-1)\) \(C(-4,1), D(-2,0), E(4,0), F(-2,-2), G(0,0),\) and \(H(0,-5)\)

Plot the points corresponding to the ordered pairs \(A(1,4), B(4,-2)\) \(C(-3,2), D(-3,0), E(3,0), F(0,0), G(-3,-3),\) and \(H(0,-2)\)

In Exercises \(7-24,\) find the general form of the equation of the line satisfying the conditions given and graph the line. Through \((-3,5)\) parallel to a line with slope \(-4\)
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