Question

Find the slope and y-intercept of each line. Graph the line. $$ x-y=2 $$

Short answer

Slope: 1, y-intercept: -2

Step-by-step solution

Rewrite the equation in slope-intercept form

To identify the slope and y-intercept, rewrite the given equation in the slope-intercept form, which is \(y = mx + b\). Start with the equation \(x - y = 2\). Isolate \(y\) by moving \(x\) to the other side of the equation: \(-y = -x + 2\). Then, multiply both sides by \(-1\) to solve for \(y\): \(y = x - 2\).

Identify the slope and y-intercept

In the equation \(y = x - 2\), compare it to the standard form \(y = mx + b\). Here, \(m = 1\) is the slope, and \(b = -2\) is the y-intercept.

Plot the y-intercept

On a graph, locate the y-intercept, which is the point where the line crosses the y-axis. For this equation, the y-intercept is \(-2\), so plot the point \( (0, -2) \).

Use the slope to find another point

From the y-intercept \( (0, -2) \), use the slope \(m = 1 \). The slope \(m = 1\) indicates a rise of 1 unit and a run of 1 unit. From the point \( (0, -2) \), move up 1 unit and to the right 1 unit to plot the next point at \( (1, -1) \).

Draw the line

Draw a straight line through the points \( (0, -2) \) and \((1, -1)\). This is the graph of the line \( y = x - 2 \).

Key concepts

identifying the slope and y-intercept

To fully understand the concept of identifying the slope and y-intercept, we start with the slope-intercept form of a linear equation, which is written as:
\(y = mx + b\)
Here:

• \(m\) represents the slope.
• \(b\) represents the y-intercept.

When an equation is given, the first step is to rewrite it in this form if it’s not already. For example, given the equation \(x - y = 2\), we isolate \(y\) to put it in slope-intercept form:

• Subtract \(x\) from both sides: \(-y = -x + 2\)
• Multiply by -1: \(y = x - 2\)

Now the equation is in slope-intercept form: \(y = x - 2\). Here, it’s easy to identify that:

• Slope (\(m\)) = 1 (since the coefficient of \(x\) is 1)
• y-intercept (\(b\)) = -2

This provides a clear pathway to graphing the linear equation.

graphing linear equations

Graphing a linear equation involves plotting points on a coordinate plane and drawing a straight line through these points. Let’s use our equation from earlier, \(y = x - 2\):

• **Plot the y-intercept**: Start by finding the y-intercept, which is -2. This means the line crosses the y-axis at the point \((0, -2)\). Plot this point.
• **Use the slope to find another point**: The slope \(m\) is 1, which means for every 1 unit you move up, you also move 1 unit to the right. Starting from \((0, -2)\), move up 1 unit and right 1 unit to get to the point \((1, -1)\). Plot this point as well.
• **Draw the line**: Use a ruler or a straight edge to draw a line through the two points, extending in both directions.

By using these steps, you will create an accurate graph of the line. Remember, the slope tells you the direction and steepness, while the y-intercept tells you where the line crosses the y-axis.

transforming equations into slope-intercept form

Transforming an equation into slope-intercept form \(y = mx + b\) is an essential step in solving many linear equations. Here’s a simple guide to transform and understand equations in this form:

• **Isolate y**: Start by moving all terms involving \(y\) to one side of the equation. For example, if you start with \(x - y = 2\), move \(x\) to the right side by subtracting it from both sides: \(-y = -x + 2\).
• **Isolate y completely**: Remove any coefficients from \(y\). In this case, multiply the whole equation by -1 to make \(y\) positive: \(y = x - 2\).

Once in slope-intercept form, the equation \(y = x - 2\) reveals the slope \(m\) (which is 1) and the y-intercept \(b\) (which is -2). This makes graphing and understanding relationships between variables simpler.
These steps form a foundation for understanding and applying linear equations in various mathematical contexts.