Problem 54 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the line. Graph the equation. Label the points where the line crosses the axes. $$ 4 x+5 y=20 $$

Step-by-Step Solution

Answer

The \(x\)-intercept of the line is at (5,0) and the \(y\)-intercept is at (0,4). These are the points where the given line crosses the axes.

Step 1: Solving for \(x\)-intercept

To find the \(x\)-intercept, set \(y=0\) and solve for \(x\) in the given equation. The equation thus becomes \(4x + 5(0) = 20\), which simplifies to \(4x = 20\). By solving this equation we find that when \(y=0\), \(x=5\). Thus, the \(x\)-intercept of the line is at the point (5,0).

Step 2: Solving for \(y\)-intercept

To find the \(y\)-intercept, set \(x=0\) and solve for \(y\) in the equation. The equation thus becomes \(4(0) + 5y = 20\), which simplifies to \(5y = 20\). Solving for \(y\) we find that when \(x=0\), \(y=4\). Thus, the \(y\)-intercept of the line is at the point (0,4).

Step 3: Graphing the equation

To graph the equation, simply plot the intercept points from step 1 and 2 on a graph: that would be (0,4) and (5,0). Then draw a line that goes through these two points. Extend the line to the edges of the graph. The line represents all the solutions of the equation.

Key Concepts

Intercepts

Finding intercepts is a key step in graphing linear equations. The intercepts are the points where the line crosses the axes in a coordinate plane. Intercepts can be easily identified, providing a solid foundation for the graph of an equation.

• X-intercept: It occurs where the line crosses the x-axis. Here, the y-coordinate is always zero. To find it, set \( y = 0 \) in the equation and solve for \( x \).
• Y-intercept: This is where the line crosses the y-axis. At this point, the x-coordinate is always zero. To find it, set \( x = 0 \) and solve for \( y \).

For instance, in the equation \( 4x + 5y = 20 \), the x-intercept is found by substituting \( y = 0 \), giving us the point (5,0). Similarly, by substituting \( x = 0 \), we find the y-intercept (0,4). These points will help immensely when it comes to graphing the equation.

Graphing

Graphing linear equations is an efficient way to visualize mathematical relationships. It involves plotting points on a Cartesian coordinate system where x and y values are labeled on respective axes. To graph an equation like \( 4x + 5y = 20 \), start by plotting the intercepts found:

• Mark the x-intercept (5,0) on the horizontal (x) axis.
• Mark the y-intercept (0,4) on the vertical (y) axis.

Next, draw a straight line through these intercepts. Extend the line across the graph, showing that it includes all possible solutions. When graphing, always label the intercepts clearly to show where the line meets the axes. This makes it easier to analyze and understand the equation from the graph.

Solving Equations

Solving equations is about finding values that satisfy a given mathematical statement. In linear equations, solving often involves isolating variables to understand relationships between them. Equations like \( 4x + 5y = 20 \) are solved by applying basic algebraic techniques to determine the intercepts or particular solutions. Here's how it works:

• To find the x-intercept, make \( y = 0 \) and solve for \( x \). This involves simplifying the equation to \( 4x = 20 \) which solves to \( x = 5 \).
• To find the y-intercept, make \( x = 0 \) and solve for \( y \). Simplifying to \( 5y = 20 \) gives \( y = 4 \).

These calculations reveal critical points for graphing and understanding the behavior of the line represented by the equation. Solving equations facilitates the discovery of patterns and solutions, allowing further exploration of mathematical models.