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Chapter 2: Problem 89

Find the slope and y-intercept of each line. Graph the line. $$ x+y=1 $$

Short Answer

Expert verified
Slope is -1, and y-intercept is 1.

Step by step solution

01

Rewrite 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 in the given equation: x + y = 1 y = -x + 1.
02

Identify the Slope

From the equation y = -x + 1, the coefficient of x is the slope. Therefore, the slope (m) is -1.
03

Identify the Y-Intercept

In the equation y = -x + 1, the constant term is the y-intercept. Therefore, the y-intercept (b) is 1.
04

Graph the Line

To graph the line, plot the y-intercept (0, 1) on the y-axis. From this point, use the slope to determine another point on the line. Since the slope is -1, move one unit down and one unit right, ending at the point (1, 0). Draw the line through these points.

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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 describe a straight line on a graph. They typically take the form of: \(ax + by = c\), but can be expressed in slope-intercept form for easier graphing.

In slope-intercept form, linear equations are written as: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
This form is particularly useful for quickly identifying key properties of the line. Understanding how to convert between standard form and slope-intercept form is essential.

For example, consider the equation \(x + y = 1\). By rearranging it to isolate \(y\), we get \(y = -x + 1\), which is now in slope-intercept form.
graphing lines
The y-intercept of a line is the point where the line crosses the y-axis.

In the equation \(y = -x + 1\), the y-intercept is 1.

This is represented by the point (0, 1) on the graph.

To visualize this, start graphing from the y-intercept.

From (0, 1), use the slope to find the next point on the line.

The slope equals the 'rise over run'.

Here, the slope is \(-1\), which means for each unit you move vertically down, move horizontally to the right.

Plot these points accurately to represent the linear relationship.

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Most popular questions from this chapter

Kinetic Energy The kinetic energy \(K\) of a moving object varies directly with its mass \(m\) and the square of its velocity \(v\). If an object weighing 25 kilograms and moving with a velocity of 10 meters per second has a kinetic energy of 1250 joules, find its kinetic energy when the velocity is 15 meters per second.

(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ -0.3 x+0.4 y=1.2 $$

Measuring Temperature The relationship between Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and Fahrenheit \(\left({ }^{\circ} \mathrm{F}\right)\) degrees of measuring temperature is linear. Find a linear equation relating \({ }^{\circ} \mathrm{C}\) and \({ }^{\circ} \mathrm{F}\) if \(0^{\circ} \mathrm{C}\) corresponds to \(32^{\circ} \mathrm{F}\) and \(100^{\circ} \mathrm{C}\) corresponds to \(212^{\circ} \mathrm{F}\). Use the equation to find the Celsius measure of \(70^{\circ} \mathrm{F}\).

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. Perpendicular to the line \(x=8\); containing the point (3,4)

Geometry Use slopes to show that the quadrilateral whose vertices are \((1,-1),(4,1),(2,2),\) and (5,4) is a parallelogram.
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