Problem 50 - 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. $$ -4 x+3 y=24 $$

Step-by-Step Solution

Answer

The x-intercept of the line is 6, and the y-intercept is 8.

Step 1: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). So,\[-4x + 3 * 0 = 24 \-4x = 24 \x = - \frac{24}{-4} = 6\]So, the x-intercept is 6.

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). So,\[-4 * 0 + 3y = 24 \3y = 24 \y = \frac{24}{3} = 8\]So, the y-intercept is 8.

Step 3: Graph the equation

Plot the line using the x-intercept (6,0) and the y-intercept (0,8). The line crossing these points represents the equation \(-4x + 3y = 24\)

Key Concepts

Graphing Equations

Graphing an equation is like drawing its path on a coordinate plane. It's an easy way to visually see the relationship between the variables in an equation. To graph a linear equation, you commonly use the x-intercept and y-intercept because these points are where the line crosses the axes.

Here's the basic process of graphing a linear equation:

• Find the x-intercept by setting y to 0 in the equation.
• Find the y-intercept by setting x to 0 in the equation.
• Plot these intercepts on the graph.
• Draw a straight line through these two points.

This line represents all the solutions to the equation. Every point on the line solves the equation when substituted back into the equation. Alleviate worries with graphing by thinking of it like connecting the dots. Just remember that a linear equation will always graph as a straight path. This method efficiently helps you identify the shape and direction the line will have on the coordinate plane.

Linear Equations

Linear equations are equations of the first degree, meaning the highest power of the variable(s) is one. They're used to describe a straight line in the coordinate plane. A typical form of a linear equation in two variables, x and y, is given as ax + by = c.

Some key features of linear equations include:

• Each solution to the equation corresponds to a point on the line.
• The graph of a linear equation is a straight line.
• A linear equation can have either one variable or two, with two variables fitting the structure ax + by = c.
• The coefficients a, b, and c structure the relationship between x and y.

Understanding linear equations helps in learning how lines behave and how they relate to each other. They also form the basics of more complex equations and mathematical concepts.

X-Intercept and Y-Intercept

Intercepts are fundamental in understanding where a line interacts with the axes on a graph. The x-intercept and y-intercept are simply the points where the line crosses the x-axis and y-axis, respectively.

Here's how you find them:

• The x-intercept occurs when y = 0. Substitute y = 0 into the equation, then solve for x. This gives you the point (x, 0).
• The y-intercept occurs when x = 0. Substitute x = 0 into the equation, then solve for y. This gives you the point (0, y).

These intercepts are immensely useful for easily sketching linear equations. Since these values show where the line crosses the axes, they provide quick and key insights into how the line behaves and how it divides the plane. They simplify the process of graphing by reducing the need for complicated calculations.