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Chapter 5: Problem 102

Find the slope and the \(y\)-intercept of the line. \(3 x-10 y=14\)

Short Answer

Expert verified
Slope: \(\frac{3}{10}\), y-intercept: \(-\frac{7}{5}\).

Step by step solution

01

Write the Equation in 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. Start by isolating \(y\) on one side of the equation.
02

Rearrange the Given Equation

Starting with the given equation: \[3x - 10y = 14\]. Subtract \(3x\) from both sides to get: \[-10y = -3x + 14\].
03

Solve for y

Divide both sides of the equation by \(-10\) to solve for \(y\): \[y = \frac{-3}{-10}x + \frac{14}{-10}\].
04

Simplify the Equation

Simplify the equation: \[y = \frac{3}{10}x - \frac{7}{5}\]. Now the equation is in slope-intercept form where the slope \(m\) is \(\frac{3}{10}\) and the y-intercept \(b\) is \(-\frac{7}{5}\).

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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 mathematical statements of the form: \[Ax + By = C\] They represent straight lines when graphed on a coordinate plane. The terms 'linear' refer to the fact that all the variables appear to the first power, giving the graph its straight-line shape. The coefficients \(A\), \(B\), and \(C\) indicate specific details about the steepness and position of the line.
slope
The slope of a line is a measure of its steepness. It indicates how much the y-coordinate changes for every one-unit change in the x-coordinate. In the slope-intercept form, the slope is represented by \(m\) in the equation \[y = mx + b\]. For example, in the equation \[y = \frac{3}{10}x - \frac{7}{5}\], the slope \(m\) is \(\frac{3}{10}\). A positive slope means the line rises as it moves from left to right, while a negative slope means the line falls. A zero slope indicates a horizontal line, and an undefined slope corresponds to a vertical line.
y-intercept
The y-intercept is the y-coordinate of the point where a line crosses the y-axis. It shows where the line intersects the y-axis and is represented by \(b\) in the slope-intercept form \(y = mx + b\). In our example equation \[y = \frac{3}{10}x - \frac{7}{5}\], the y-intercept \(b\) is \(-\frac{7}{5}\). To find the y-intercept, set \(x\) to zero and solve for \(y\). This gives the specific point where the line crosses the y-axis.
simplification
Simplification is the process of making an equation easier to understand and work with. This step usually involves combining like terms, reducing fractions, and writing the equation in a standard form. Starting with the equation \[3x - 10y = 14\], we rearranged and simplified it to the slope-intercept form \[y = \frac{3}{10}x - \frac{7}{5}\]. Simplification allows us to easily identify the slope and y-intercept, making it straightforward to graph the line or use it in further calculations.

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Most popular questions from this chapter

U.S. Imports. The amount of imports to the United States has increased exponentially since 1980 (Sources: U.S. Census Bureau; U.S. Bureau of Economic Analysis; U.S. Department of Commerce). The exponential function $$ I(x)=297.539(1.075)^{x} $$ where \(x\) is the number of years after \(1980,\) can be used to estimate the total amount of U.S. imports, in billions of dollars. Find the total amount of imports to the United States in \(1995,\) in \(2005,\) in \(2010,\) and in \(2013 .\) Round to the nearest billion dollars. (IMAGE CANT COPY)

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\log _{5}(x+4)+\log _{5}(x-4)=2$$

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\log _{a} x^{3}=3 \log _{a} x$$

Use a graphing calculator to find the approximate solutions of the equation. $$4 x-3^{x}=-6$$

Solve using any method. Given that \(a=\log _{8} 225\) and \(b=\log _{2} 15,\) express as a function of \(b\).
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