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Chapter 5: Problem 14

Write an equation of the line in slope-intercept form. The slope is \(0 ;\) the \(y\) -intercept is 4

Short Answer

Expert verified
The equation of the line in slope-intercept form is \(y=4\).

Step by step solution

01

Understand the slope-intercept form

The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

Plugging in the slope

The slope given in the problem is 0. Hence, \(m = 0\) in the equation.
03

Plugging in the y-intercept

The y-intercept given is 4. Hence, \(b = 4\) in the equation.
04

Substitute the slope and y-intercept into the slope-intercept form

Finally, substitute the slope and y-intercept into the equation \(y = mx + b\) to get the equation of the line in slope intercept form: \(y=0x+4\).

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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 fundamental in algebra and represent straight lines on a graph. The general form of a linear equation is expressed as \(y = mx + b\), known as the slope-intercept form.
This form provides a straightforward way to identify the slope and y-intercept of the line, making it easier to visualize and analyze.
Unlike other types of equations, linear equations result in a straight line when plotted on a coordinate plane.
  • The graph never curves; it always forms a straight line.
  • It鈥檚 defined for every real number, allowing for endless possibilities along the line.
Whenever you see a linear equation, you can quickly determine these key aspects without more complex calculations, thanks to the simplified structure of \(y = mx + b\).
Slope
The slope of a line, represented as \(m\) in the slope-intercept form \(y = mx + b\), indicates how steep the line is. It shows the rate of change between the vertical and horizontal components of the line.
If a line has a slope of 0, it means that the line is perfectly horizontal. The y-values remain constant no matter what value x takes.
  • A positive slope means the line is rising as you move from left to right.
  • A negative slope indicates the line is falling as you move from left to right.
To calculate the slope from two points on a line, use the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
This formula finds the difference between the y-values and the x-values of two points on the graph.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the equation \(y = mx + b\), \(b\) represents the y-intercept. The unique feature of the y-intercept is that it occurs when \(x = 0\).
This point is crucial because:
  • It provides an easy reference point when graphing a line.
  • It helps in determining the starting point of a line on a graph.
In linear equations, knowing the y-intercept allows for quick plotting of the line without extensive calculations.
For example, in our exercise, the y-intercept is 4, meaning the line crosses the y-axis at the point (0,4). This simple information is often all you need alongside the slope to graph a line.
Algebra
Algebra is a branch of mathematics that involves expressions with variables.
It's all about finding unknown values by using symbols and letters to represent numbers. Linear equations are a foundational concept in algebra, where relationships between variables are expressed.
  • Algebra helps in solving problems by developing equations or formulas.
  • It makes use of operations like addition, subtraction, multiplication, and division applied to variables.
In our example of a linear equation \(y = 0x + 4\), algebra enabled us to express a relationship where the y-values are constant, further emphasizing its ability to simplify and solve real-world problems.
Algebra aids in understanding patterns and constructing strategies for finding unknown quantities, a critical skill in various fields of science and engineering.

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Most popular questions from this chapter

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. \(-7\) and 2

The time \(t\) (in hours) needed to produce \(x\) units of a product is modeled by \(t=p x+s .\) If it takes 265 hours to produce 200 units and 390 hours to produce 300 units, what is the value of \(s ?\) (A) 1.25 (B) 15 (C) 100 (D) 125

Graph the line that passes through the points. Write its equation in slope- intercept form. (Review \(5.3 \text { for } 5.7)\) $$(14,-9),(-5,-3)$$

Write an equation in standard form of the line that passes through the two points. $$(-4,1),(2,-5)$$

Write an equation of the line in slope-intercept form. The slope is \(\frac{1}{2} ;\) the \(y\) -intercept is \(-8\).
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