Problem 57 - Algebra 2

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of each equation. Then graph the equation. $$ -2 x+5 y=20 $$

Step-by-Step Solution

Answer

X-intercept: (-10, 0); Y-intercept: (0, 4); Graph: Straight line through intercepts.

Step 1: Understand the Equation

The given equation is \(-2x + 5y = 20\).This is a linear equation in two variables, which represents a straight line when graphed on a coordinate plane.

Step 2: Find the X-Intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\).\[-2x + 5(0) = 20\]\[-2x = 20\]Divide both sides by \(-2\):\[x = -10\]The x-intercept is \((-10, 0)\).

Step 3: Find the Y-Intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\).\[-2(0) + 5y = 20\]\[5y = 20\]Divide both sides by 5:\[y = 4\]The y-intercept is \((0, 4)\).

Step 4: Graph the Equation

To graph the equation, plot the intercepts from Step 2 and Step 3 on a coordinate plane. The x-intercept is at \((-10, 0)\) and the y-intercept at \((0, 4)\).Draw a straight line through these two points to represent the equation \(-2x + 5y = 20\).

Key Concepts

Understanding the X-Intercept

The x-intercept of a linear equation is the point where the line crosses the x-axis. This means that at this point, the value of y is zero. To find the x-intercept for any linear equation, you simply need to set the y-variable to zero and solve the resulting equation for x.
This method works because crossing the x-axis means the vertical position is zero, hence y = 0.
For example, in the equation \(-2x + 5y = 20\), if we set \(y = 0\), the equation becomes \(-2x = 20\). Solving for x gives \(x = -10\). Therefore, the x-intercept is at the point \((-10, 0)\).

• Remember: At the x-intercept, y is always zero.
• Use substitution to find the x-value.

This point is particularly useful in graphing, as it provides one fixed point through which the line passes.

Understanding the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. At this point, the value of x is zero. Finding the y-intercept involves setting the x-variable to zero and solving the equation for y.
This makes sense because the point of crossing the y-axis has no horizontal movement, so x = 0.
Taking the equation \(-2x + 5y = 20\) as an example and substituting \(x = 0\), we have the equation \(5y = 20\). Solving for y gives \(y = 4\). Hence, the y-intercept is at the point \((0, 4)\).

• Remember: At the y-intercept, x is always zero.
• Substitute to find the y-value.

The y-intercept is another crucial point that helps in graphing the line on a graph.

Graphing Linear Equations

Graphing a linear equation involves plotting points and connecting them to show the accurate representation of the equation on a graph. The easiest way to graph a linear equation is by finding and plotting the x-intercept and y-intercept. These points provide a reliable guide to draw the straight line that represents the equation.
In our example, after determining the intercepts, x-intercept \((-10, 0)\) and y-intercept \((0, 4)\), you plot these points on a coordinate plane.
Once plotted, draw a straight line through these two points. This line encompasses every solution of the equation \(-2x + 5y = 20\).

• Always start by plotting the intercepts.
• Use a ruler to ensure the line is straight.

By drawing a straight line, you visualize all solutions to the equation, making understanding of the linear relationship easier.