Question

Find the slope and y-intercept of each line. Graph the line. $$ y-x=0 $$

Short answer

Slope = 1, y-intercept = 0.

Step-by-step solution

- Rewrite the Equation in Slope-Intercept Form

The general slope-intercept form of a linear equation is onumber y = mx + b onumber, where m is the slope and b is the y-intercept. Rewrite the given equation onumber y - x = 0 onumber in this form. Add x to both sides of the equation: onumber y = x onumber

- Identify the Slope

Compare the equation eq y = x onumber to the slope-intercept form eq y = mx + b onumber. Here, m (the coefficient of x) is 1. Therefore, the slope (m) is 1.

- Identify the Y-intercept

From the equation eq y = x onumber, it is clear the constant term b is 0. Therefore, the y-intercept (b) is 0.

- Graph the Line

To graph the line, start at the y-intercept (0,0). Since the slope is 1, move one unit up and one unit to the right to plot another point (1,1). Draw a straight line through these points.

Key concepts

Linear Equations

Linear equations are fundamental in algebra. They describe a straight line when graphed on a coordinate plane. They have the general form:
\[ y = mx + b \]

• y: The dependent variable.
• m: The slope of the line.
• x: The independent variable.
• b: The y-intercept where the line crosses the y-axis.

Understanding linear equations is critical because they are used to model relationships between quantities. These relationships appear in various fields such as physics, economics, and biology.

Slope

The slope of a line indicates its steepness and direction. It is represented by the letter 'm' in the linear equation. Here are the key points to remember:

• Positive Slope: The line rises from left to right.
• Negative Slope: The line falls from left to right.
• Zero Slope: The line is horizontal.
• Undefined Slope: The line is vertical.

The slope is calculated as the change in y divided by the change in x:
\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]
In our problem, the equation is \[ y = x \]. Here, the slope \[ m \] is 1, indicating a line with a 45-degree angle rising from left to right.

Y-Intercept

The y-intercept is where the line crosses the y-axis. This happens when the value of x is zero. It is represented by 'b' in the linear equation.

• If \[ b \] is positive, the line crosses the y-axis above the origin.
• If \[ b \] is negative, the line crosses below the origin.
• If \[ b \] is zero, the line passes through the origin.

In the given equation \[ y = x \], \[ b = 0 \], meaning the line goes through the point (0,0).

Graphing Lines

Graphing a line involves plotting it on the Cartesian plane using its slope and y-intercept. Here is a step-by-step guide:

• Identify the y-intercept (\[ b \]). Plot this point on the y-axis.
• Use the slope (\[ m \]) to find another point. For a slope of 1, from the intercept, move up 1 unit and right 1 unit.
• Draw a straight line through the two points.

For the equation \[ y = x \], start at (0,0). Move to (1,1) by going up 1 unit and to the right 1 unit. Connect these points to form the line.