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Chapter 3: Problem 21

Write the equation of each line with the given slope and \(y\) -intercept. $$ m=4,(0,-3) $$

Short Answer

Expert verified
y = 4x - 3

Step by step solution

01

Recall the slope-intercept form

The slope-intercept form of a line is given by the equation y = mx + b where m is the slope and b is the y-intercept.
02

Identify given values

From the problem, the slope ( m ) is 4, and the y-intercept ( b ) is -3.
03

Substitute values into the slope-intercept form

Place the identified slope and y-intercept values into the slope-intercept form equation: y = 4x - 3

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Key Concepts

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

linear equations
Linear equations are equations that make straight lines when graphed. You encounter these often in algebra. The standard form is usually expressed as \[y = mx + b\].
This form is easy to work with because it immediately tells you two important things:
  • The slope of the line
  • The y-intercept
The slope indicates how steep the line is, and the y-intercept shows where the line crosses the y-axis. Let鈥檚 break these pieces down further.
slope
The slope, represented as \(m\), describes the steepness or incline of the line. In the equation \[y = mx + b\], the slope is the number multiplied by \(x\).
For the problem at hand, the slope (\(m\)) is 4.
  • If the slope is positive, the line rises as it moves to the right.
  • If the slope is negative, the line falls as it moves to the right.
A slope of 4 means that for every unit increase in \(x\), the value of \(y\) increases by 4. This steep line moves upwards quite quickly.
y-intercept
The y-intercept is where the line crosses the y-axis. It is represented as \(b\) in the slope-intercept equation \[y = mx + b\].
In our problem, the y-intercept (\(b\)) is -3.
This means when \(x\) is 0, \(y\) is -3. Therefore, the line crosses the y-axis at the point (0, -3). Knowing the y-intercept helps you quickly locate where the line starts on the graph.
algebra basics
Understanding the basics of algebra makes working with linear equations easier. You鈥檒l often use operations like addition, subtraction, multiplication, and division to solve equations and understand concepts.
Here鈥檚 a quick recap of what you need:
  • Identify the slope (\(m\)) and y-intercept (\(b\)).
  • Substitute those values into the slope-intercept form \[y = mx + b\].

In our example, we identified the slope as 4 and the y-intercept as -3. Substituting these values gives us the equation \[y = 4x - 3\]. This is your linear equation. Now you鈥檙e ready to graph or analyze it!

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Most popular questions from this chapter

Concept Check Plot each set of points, and draw a line through them. Then give the equation of the line. $$ (-5,5),(0,5), \text { and }(3,5) $$

Concept Check Plot each set of points, and draw a line through them. Then give the equation of the line. $$(3,5),(3,0), \text { and }(3,-3)$$

Write an equation of the line satisfying the given conditions. Give the final answer in slope intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines in Section \(3.3 .)\) Through \((2,-3) ;\) parallel to \(3 x=4 y+5\)

Solve each problem. The table lists the average annual cost (in dollars) of tuition and fees at 2 -year colleges for selected years, where year 1 represents \(2004,\) year 2 represents \(2005,\) and so on. \(\begin{array}{|c|c|}\hline \text { Year } & {\text { cost }(\text { in dollars })} \\ {1} & {2079} \\ {2} & {2182} \\ {3} & {2272} \\ {4} & {2361} \\ {5} & {2402} \\ \hline\end{array}\) (a) Write five ordered pairs from the data. (b) Plot the ordered pairs. Do the points lie approximately in a straight line? (c) Use the ordered pairs \((1,2079)\) and \((4,2361)\) to write an equation of a line that approximates the data. Give the final equation in slope-intercept form. (d) Use the equation from part (c) to estimate the average annual cost at 2 -year colleges in \(2009 \text { to the nearest dollar. (Hint: What is the value of } x \text { for } 2009 ?)\)

Graph each equation by using the slope and y-intercept. $$ 6 x-5 y=30 $$
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