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Chapter 7: Problem 16

Graph the equations. $$ y=x $$

Short Answer

Expert verified
Answer: The slope of the line represented by the equation \(y=x\) is 1, and its angle with the x-axis is \(45^\circ\).

Step by step solution

01

Identify the slope and y-intercept

The given equation is in the form \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. In our case, \(m=1\) and \(b=0\).
02

Plot the y-intercept

Since the y-intercept is \(0\), we will plot a point at the origin \((0,0)\).
03

Use the slope to find another point

Since the slope \(m=1\), it means that the line rises by 1 unit for every 1 unit it moves to the right. So from the point \((0,0)\), we can move one unit to the right and one unit up to reach the point \((1,1)\). We'll plot this point as well.
04

Draw the line

Now that we have two points \((0,0)\) and \((1,1)\) on the graph, we can draw a straight line passing through these points. That's it! You've successfully graphed the equation \(y=x\). The line passes through the origin and has a slope of 1, which means it makes a \(45^\circ\) angle with the x-axis.

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form of a linear equation is foundational for graphing lines. It is expressed as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) signifies the y-intercept.

Let's break it down using the given exercise equation \(y = x\). Here, this equation can be seen as \(y = 1x + 0\), which clearly shows the slope \(m = 1\) and the y-intercept \(b = 0\). This form allows us to quickly identify key characteristics of the line without needing to perform complex calculations or plot numerous points.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. This is vital because it provides a starting point for graphing the line. In our exercise, because the y-intercept \(b\) is 0, the graph begins at the origin \((0,0)\).

Remember, the y-intercept is always where the line intersects the y-axis, and its x-coordinate will always be zero. No matter how steep or flat the line, if you follow it till it meets the y-axis, that's the y-intercept.
Slope
The slope is a number that describes both the direction and the steepness of the line. A positive slope, as seen in our exercise with \(m=1\), means that as we move from left to right, the line rises. Conversely, a negative slope indicates that the line falls.

In our step by step solution, we moved one unit to the right and one unit up from the y-intercept to follow the slope. This 'rise over run' method is a great way to visualize and graph any linear equation once you know the slope.
Coordinate Graphing
Coordinate graphing is the process of placing points on a graph based on their x and y coordinates and forming shapes, such as lines. In our exercise, plotting the points \((0,0)\) and \((1,1)\) and drawing a line through them completed the graphing of the linear equation.

Remember that every point on the coordinate plane has a unique place based on its x (horizontal) and y (vertical) values. Mastering this technique allows for the representation of equations visually, making it easier to understand their behavior and relationship to other mathematical concepts.

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Most popular questions from this chapter

Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ m=-1, \quad(2,0) $$

Determine the slope and \(y\) -intercept of the lines. $$ y=-x+2 $$

Determine the slope and \(y\) -intercept of the lines. $$ 5 y+4 x=6 $$

Determine the slope and \(y\) -intercept of the lines. $$ y=9 x-1 $$

Graph the inequalities. $$ y \geq-2 $$
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