Question

Find the \(x\) -intercept and the \(y\) -intercept of the line. Graph the equation. Label the points where the line crosses the axes. $$ 2 x+9 y=-36 $$

Short answer

The x-intercept of the line is at (-18, 0) and the y-intercept of the line is at (0, -4). When graphed, the line should cross through these two points.

Step-by-step solution

Find the x-intercept

To find the x-intercept, set \(y\) to 0. Substituting \(y = 0\) we get: \(2x + 9(0) = -36\), which simplifies to \(2x = -36\). Solving for \(x\), we divide both sides by 2: \(x = -36 / 2\), which simplifies to \(x = -18\). Therefore, the x-intercept is at (-18, 0).

Find the y-intercept

The y-intercept is found by setting \(x\) to zero. Substituting \(x = 0\) in the equation: \(2(0) + 9y = -36\), which simplifies to \(9y = -36\). Solving for \(y\), we divide both sides by 9: \(y = -36 / 9\), which simplifies to \(y = -4\). Therefore, the y-intercept is at (0, -4).

Graph the line

Now, plot the x-intercept and y-intercept on a graph. Draw a line through these two points. The line crosses the x-axis at (-18, 0) and the y-axis at (0, -4)

Key concepts

X-Intercept

The x-intercept of a line on a graph is the point where the line crosses the x-axis. This is where the value of y is zero. It tells you how far along the x-axis the line will cross. In solving for the x-intercept, you essentially remove the y variable by setting it to zero, and then solve the resulting equation for x. For the equation 2x + 9y = -36, setting y equal to 0 and solving for x, as shown in the provided steps, gives us an x-intercept at (-18, 0).

This point is crucial for graphing linear equations because once you have it, you have found one of the two points needed to draw the line. Remember, it's always coupled with a y-coordinate of zero and falls along the x-axis.

Y-Intercept

In contrast, the y-intercept is where the line crosses the y-axis, which occurs when the x value is zero. To find it, we take the original linear equation and set the x variable to zero. Continuing from the solution steps above, substituting x with 0 in the equation 2x + 9y = -36 and solving for y, we arrive at the y-intercept (0, -4).

The y-intercept gives us the starting point of the line on the graph. Just like the x-intercept, it is essential because it represents one of the two key points that define a linear equation's graph. The coordinate will always include an x-value of zero and appear on the y-axis.

Solving Linear Equations

Solving linear equations involves finding the values of variables that make the equation true. Linear equations can be simplified and rearranged to find the solutions for the unknowns, typically represented as x and y in two dimensions. In our example, to find the intercepts, we solved the linear equation by isolating one variable at a time, setting the other variable to zero.

This process of solving is fundamental: it includes operations like addition, subtraction, multiplication, and division to manipulate the equation — until we get the variable isolated on one side of the equals sign. Identifying the x and y intercepts is often one of the first steps when graphing a linear equation, because it simplifies the equation into terms that can be easily plotted on a graph.

Plotting Points

After solving for the x and y intercepts, the next step is plotting points on the graph. To plot a point, you find the corresponding location on the graph's grid and mark it. For the equation we’ve been working with, the points (-18, 0) for the x-intercept and (0, -4) for the y-intercept would be plotted.

Once the points are plotted, draw a straight line through these points, extending it to the edges of your graph. This line represents all the solutions to the equation. Placing a point on the graph is essentially drawing a dot where the x and y coordinates meet. Remember, accuracy is important when plotting points as it ensures the graph's representation of the equation is correct.