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Magazine Chapter 2: Problem 90
Find the slope and y-intercept of each line. Graph the line. $$ x-y=2 $$
Short Answer
Expert verified
Slope: 1, y-intercept: -2
Step by step solution
01
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 \(y\): \(y = x - 2\).
02
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.
03
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 \( (0, -2) \).
04
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 \( (1, -1) \).
05
Draw the line
Draw a straight line through the points \( (0, -2) \) and \((1, -1)\). This is the graph of the line \( y = x - 2 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
When an equation is given, the first step is to rewrite it in this form if it鈥檚 not already. For example, given the equation \(x - y = 2\), we isolate \(y\) to put it in slope-intercept form:
Now the equation is in slope-intercept form: \(y = x - 2\). Here, it鈥檚 easy to identify that:
This provides a clear pathway to graphing the linear equation.
\(y = mx + b\)
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鈥檚 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鈥檚 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鈥檚 use our equation from earlier, \(y = x - 2\):
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.
- **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鈥檚 a simple guide to transform and understand equations in this form:
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.
- **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.
