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Chapter 7: Problem 115

Find the \(x\) -intercept and the \(y\) -intercept of the line given by \(x-y=10\)

Short Answer

Expert verified
The y-intercept is (0, -10) and the x-intercept is (10, 0).

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. Substitute x = 0 into the equation so it becomes: \[ 0 - y = 10 \] Simplify to find: \[ -y = 10 \] Divide both sides by -1 to get: \[ y = -10 \] Therefore, the y-intercept is (0, -10).
02

- Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. Substitute y = 0 into the equation so it becomes: \[ x - 0 = 10 \] Simplify to find: \[ x = 10 \] Therefore, the x-intercept is (10, 0).

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Key Concepts

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

x-intercept
The x-intercept of a line is the point where the line crosses the x-axis. To find the x-intercept, you need to set the value of y to 0 in the given equation and then solve for x.
For example, let鈥檚 take the equation from our exercise: \[ x - y = 10 \] To find the x-intercept, replace y with 0: \[ x - 0 = 10 \] Simplify the equation: \[ x = 10 \] So, the x-intercept is at the point (10, 0). This means the line crosses the x-axis at 10. Remember, at any x-intercept, the value of y is always 0.
y-intercept
The y-intercept is the point where the line crosses the y-axis. To find the y-intercept, you need to set the value of x to 0 in the given equation and solve for y.
Using the same equation from our exercise: \[ x - y = 10 \] Set x to 0: \[ 0 - y = 10 \] Simplify to find y: \[ -y = 10 \] Divide both sides by -1: \[ y = -10 \] Therefore, the y-intercept is (0, -10). This means the line crosses the y-axis at -10. At any y-intercept, the value of x is always 0.
linear equations
Linear equations are equations of the first degree, which means they have no variables raised to a power higher than one. The general form of a linear equation in two variables is: \[ Ax + By = C \] where A, B, and C are constants. This form is often called the standard form of a linear equation.

Key characteristics of linear equations include:
  • They graph as straight lines.
  • Their solutions are any points (x, y) that lie on the line.
  • The slope (steepness) of the line is given by the ratio -A/B when B is not zero.
  • They have at most one y-intercept and one x-intercept.
In our exercise, the equation: \[ x - y = 10 \] is a linear equation where A = 1, B = -1, and C = 10. It helps us find the relationship between x and y, so we can determine points like intercepts by setting one variable to zero and solving for the other.

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