Problem 49
Question
Find the \(x\) - and \(y\) -intercepts for each line and use them to graph the line. $$x-y+5=0$$
Step-by-Step Solution
Verified Answer
x-intercept: (-5, 0), y-intercept: (0, 5).
1Step 1 - Understanding the Equation
Given the linear equation in the form: \(x - y + 5 = 0\).
2Step 2 - Find the x-intercept
To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\): \[x - (0) + 5 = 0\]\[x + 5 = 0\]\[x = -5\]So, the x-intercept is \((-5, 0)\).
3Step 3 - Find the y-intercept
To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\): \[(0) - y + 5 = 0\]\[-y + 5 = 0\]\[-y = -5\]\[y = 5\]So, the y-intercept is \((0, 5)\).
4Step 4 - Graph the Line
Use the intercepts to graph the line:1. Plot the x-intercept \((-5, 0)\) on the Cartesian plane.2. Plot the y-intercept \((0, 5)\) on the Cartesian plane.3. Draw a straight line through these two points, extending it in both directions.
Key Concepts
x-intercepty-interceptCartesian plane
x-intercept
The x-intercept is the point where a line crosses the x-axis of a graph. This means that at the x-intercept, the value of y is always zero.
To find the x-intercept for a given equation, simply set y to 0 and solve for x.
For example, in the equation \(x - y + 5 = 0\), setting \(y = 0\) gives:
On a graph, this means the point that touches the x-axis at \(x = -5\).
To find the x-intercept for a given equation, simply set y to 0 and solve for x.
For example, in the equation \(x - y + 5 = 0\), setting \(y = 0\) gives:
- \(x + 5 = 0\)
- Solving for x, we get \(x = -5\)
On a graph, this means the point that touches the x-axis at \(x = -5\).
y-intercept
The y-intercept is the point where a line crosses the y-axis of a graph. At this point, the value of x is always zero.
To find the y-intercept for a given equation, simply set x to 0 and solve for y.
For instance, in the equation \(x - y + 5 = 0\), setting \(x = 0\) gives:
This means the point that touches the y-axis at \(y = 5\).
To find the y-intercept for a given equation, simply set x to 0 and solve for y.
For instance, in the equation \(x - y + 5 = 0\), setting \(x = 0\) gives:
- \(-y + 5 = 0\)
- Solving for y, we get \(y = 5\)
This means the point that touches the y-axis at \(y = 5\).
Cartesian plane
The Cartesian plane is a two-dimensional graphing system that helps us plot points, lines, and curves. The plane is divided into four quadrants by the x-axis and the y-axis.
The x-axis runs horizontally, while the y-axis runs vertically. Each point on the plane is represented as \((x, y)\), where x is the horizontal distance from the origin and y is the vertical distance.
When graphing a line, finding the x- and y-intercepts helps place it correctly on this plane.
For example, to graph the line given by the equation \(x - y + 5 = 0\), we:
The x-axis runs horizontally, while the y-axis runs vertically. Each point on the plane is represented as \((x, y)\), where x is the horizontal distance from the origin and y is the vertical distance.
When graphing a line, finding the x- and y-intercepts helps place it correctly on this plane.
For example, to graph the line given by the equation \(x - y + 5 = 0\), we:
- Find the intercepts \((-5, 0)\) and \((0, 5)\)
- Plot these points on the Cartesian plane
- Draw a line through these points
Other exercises in this chapter
Problem 47
In each case find the slope of line I and graph both lines that are mentioned. Line \(l\) is perpendicular to a line with slope \(\frac{4}{5}\). Both lines cont
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View solution