Problem 9 - Understanding Elementary Algebra with Geometry

Question

Find the \(x\) - and \(y\) -intercepts of the equation. $$x-y=7$$

Step-by-Step Solution

Answer

The x-intercept is (7, 0) and the y-intercept is (0, -7).

Step 1: Identify the x-intercept

To find the x-intercept, set y to 0 and solve for x. Substitute y = 0 into the equation.

Step 2: Solve for x with y = 0

Substitute y = 0 into the equation: \(x - 0 = 7\). Simplify to find \(x = 7\). Thus, the x-intercept is at the point (7, 0).

Step 3: Identify the y-intercept

To find the y-intercept, set x to 0 and solve for y. Substitute x = 0 into the equation.

Step 4: Solve for y with x = 0

Substitute x = 0 into the equation: \(0 - y = 7\). Simplify to find \(y = -7\). Thus, the y-intercept is at the point (0, -7).

Key Concepts

x-intercept

The x-intercept is the point where a line crosses the x-axis on a graph. To find it, we set the y-value to 0 in the equation. In this case, the equation is

\( x - y = 7 \).

Substituting y = 0, the equation becomes

\( x - 0 = 7 \).

Simplifying, we find \( x = 7 \).

Therefore, the x-intercept is at the coordinate point (7, 0).

It's the point along the x-axis where the line passes through.

y-intercept

The y-intercept is where the line crosses the y-axis, which means the x-value is 0.

Given the equation

\( x - y = 7 \)

To find the y-intercept, we substitute x = 0:

\( 0 - y = 7 \).

Simplifying, we get \( -y = 7 \).

To solve for y, we divide by -1, yielding \( y = -7 \).

Thus, the y-intercept is at the coordinate point (0, -7).

This is the point where the line crosses the y-axis.

solving linear equations

Solving linear equations involves finding the values of variables that make the equation true.

For the equation

\( x - y = 7 \),

you can find specific values for x and y by isolating one variable at a time.

For x-intercept: set y to 0 and solve for x.

For y-intercept: set x to 0 and solve for y.

This method helps in quickly finding intercepts, which are crucial in graphing.

coordinate points

Coordinate points represent a location on a graph with an x-value and a y-value, written as (x, y).

Each point tells you exactly where to place a dot on the Cartesian plane.

For example, the coordinates for the x-intercept are (7, 0), meaning move 7 units along the x-axis.

The y-intercept coordinates are (0, -7), meaning move down 7 units on the y-axis.

Knowing how to read and plot coordinate points is essential for graphing linear equations.