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Chapter 2: Problem 21

Write an equation in slope-intercept form of the line that satisfies the given conditions. See Example 1. $$ \text { Slope } \frac{2}{5} ; y \text { -intercept }(0,5) $$

Short Answer

Expert verified
y = \(\frac{2}{5}\)x + 5.

Step by step solution

01

Identify the Slope-Intercept Form

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

Substitute the Given Slope

In this exercise, the slope (m) is given as \(\frac{2}{5}\). Substitute this value into the equation: y = \(\frac{2}{5}\)x + b.
03

Substitute the Given Y-intercept

The y-intercept (b) is given as 5. Substitute this value into the equation: y = \(\frac{2}{5}\)x + 5.
04

Write the Final Equation

Combine the slope and the y-intercept into the final equation: y = \(\frac{2}{5}\)x + 5.

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Key Concepts

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

linear equation
A linear equation represents a straight line on a graph. It has the general form of y = mx + b, where y is the dependent variable, x is the independent variable, m is the slope, and b is the y-intercept. Linear equations are fundamental in algebra, as they describe relationships between variables that change at a constant rate. They are used to model real-life situations, making them highly valuable in various fields such as physics, engineering, and economics. When graphing a linear equation, the slope determines the steepness of the line, while the y-intercept indicates where the line crosses the y-axis.
slope
The slope of a line, represented by m in a linear equation, measures the line's steepness and direction. It indicates how much y changes for a given change in x. Mathematically, slope is defined as the ratio of the vertical change (rise) to the horizontal change (run). In the equation y = mx + b, if m is positive, the line slopes upward from left to right. If m is negative, the line slopes downward. For example, in the exercise, the slope is given as \(\frac{2}{5}\), meaning for every 5 units the line moves horizontally to the right, it moves 2 units up.
y-intercept
The y-intercept, represented by b in the equation y = mx + b, is the point where the line crosses the y-axis. This is the value of y when x equals zero. It gives you a starting point for graphing the line. In the given problem, the y-intercept is 5, which means the line crosses the y-axis at the point (0, 5). By knowing the y-intercept and the slope, you can easily graph a line or understand the relationship between variables in the equation.
equation writing
Writing equations in slope-intercept form involves using the given slope and y-intercept. First, identify the slope (m) and the y-intercept (b). Then substitute these values into the general form y = mx + b. For example, with a slope of \(\frac{2}{5}\) and a y-intercept of 5, the equation becomes y = \(\frac{2}{5}\)x + 5. This method is systematic, ensuring that any linear relationship can be expressed in a standard way. Practicing these steps will make writing linear equations straightforward and intuitive.
algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It allows us to represent and solve problems involving unknown values. Linear equations are a vital part of algebra, helping to describe and predict various real-world situations. In algebra, understanding the slope-intercept form of a linear equation is crucial. It provides a clear and concise way to express relationships between variables and opens up more advanced study areas like calculus, where rates of change are essential.

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Most popular questions from this chapter

Graph each line passing through the given point and having the given slope. (0,0)\(; m=\frac{5}{3}\)

Find the slope of each line, and sketch its graph. \(y=-5\)

A taxicab driver charges \(\$ 2.50\) per mile. (a) Fill in the table with the correct response for the price \(f(x)\) the driver charges for a trip of \(x\) miles. (b) The linear function that gives a rule for the amount charged is \(f(x)=\) (c) Graph this function for the domain \\{0,1,2,3\\} using the set of axes at the right. $$ \begin{array}{c|c} x & f(x) \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline 3 & \\ \hline \end{array} $$

Can the graph of a linear function have an undefined slope? Explain.

Three points that lie on the same straight line are said to be collinear. Consider the points \(A(3,1), B(6,2),\) and \(C(9,3) .\) Use the slope formula to determine whether the points \((1,-2),(3,-1),\) and (5,0) are collinear.
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