Question

(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ 0.2 x-0.5 y=1 $$

Short answer

(5,0) and (0,-2)

Step-by-step solution

Find the x-intercept

To find the x-intercept, set \( y = 0 \) and solve for \( x \). Start with the equation: \[ 0.2x - 0.5y = 1 \] Set \( y = 0 \): \[ 0.2x - 0.5(0) = 1 \] Simplify: \[ 0.2x = 1 \] Solve for \( x \): \[ x = \frac{1}{0.2} = 5 \] Thus, the x-intercept is (5, 0).

Find the y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \( y \). Start with the equation: \[ 0.2x - 0.5y = 1 \] Set \( x = 0 \): \[ 0.2(0) - 0.5y = 1 \] Simplify: \[ -0.5y = 1 \] Solve for \( y \): \[ y = \frac{1}{-0.5} = -2 \] Thus, the y-intercept is (0, -2).

Plot the intercepts

Plot the x-intercept (5, 0) and the y-intercept (0, -2) on the graph.

Draw the line

Draw a straight line through the points (5, 0) and (0, -2) to graph the equation \( 0.2x - 0.5y = 1 \).

Key concepts

x-intercept

To find the x-intercept of a linear equation, you set the value of y to zero and then solve the equation for x. The x-intercept represents the point where the line crosses the x-axis.
If we take the given equation \(0.2x - 0.5y = 1\), and set y to zero, we get:

• Start with \(0.2x - 0.5(0) = 1\)
• Simplify to \(0.2x = 1\)
• Solve for x by dividing both sides by 0.2 to get \(x = 5\)

This means the x-intercept is at (5, 0). It's the point where the line touches and crosses the x-axis at x = 5.

y-intercept

To find the y-intercept of a linear equation, you set the value of x to zero and solve for y. The y-intercept is where the line crosses the y-axis. For the equation \(0.2x - 0.5y = 1\), we set x to zero:

• Start with \(0.2(0) - 0.5y = 1\)
• Simplify to \(-0.5y = 1\)
• Solve for y by dividing both sides by -0.5 to get \(y = -2\)

This means the y-intercept is at (0, -2). This point indicates where the line crosses the y-axis at y = -2.

graphing linear equations

Graphing linear equations involves drawing a straight line that represents all the solutions of the equation on a coordinate plane. To graph \(0.2x - 0.5y = 1\), follow these steps:

• Find the intercepts: x-intercept is (5, 0) and y-intercept is (0, -2).
• Plot these intercept points on the graph.
• Draw a straight line through the points (5, 0) and (0, -2).

This line represents every possible pair (x, y) that satisfies the equation. By identifying the intercepts and drawing the line, you visualize the relationship described by the equation on the graph.

solving linear equations

Solving linear equations involves finding values of variables that make the equation true. In this example, solving for x and y-intercepts helps us understand the solutions.
To solve \(0.2x - 0.5y = 1\):

• Set y to 0 and solve for x to find the x-intercept.
• Set x to 0 and solve for y to find the y-intercept.

These solutions give us specific points on the graph. In practical terms, solving the equation tells us where the line crosses the x and y axes. It's a crucial step in understanding the behavior of the linear function. By solving the equation, we can study its geometric representation and further understand its solutions in context.