Question

Find the \(x\) -intercept and the \(y\) -intercept of the line. Graph the equation. Label the points where the line crosses the axes. $$ 6 x-y=36 $$

Short answer

The x-intercept of the line is at (6,0) and the y-intercept is at (0,-36). The line crosses the x-axis at x = 6 and the y-axis at y = -36.

Step-by-step solution

Find the x-intercept

The x-intercept is found by setting \(y = 0\) in the given equation and then solve for \(x\). Substituting \(y = 0\) into the equation \(6x - y = 36\), we get \(6x = 36\). Now, to solve for \(x\), divide both sides by 6. This gives \(x = 6\). So, the x-intercept of the line is at the point (6, 0).

Find the y-intercept

The y-intercept is found by setting \(x = 0\) in the given equation and then solve for \(y\). Substituting \(x = 0\) into the equation \(6x - y = 36\), we get \(-y = 36\). Now, to solve for \(y\), multiply by -1 on both sides. This gives \(y = -36\). So, the y-intercept of the line is at the point (0, -36).

Graph the Equation

To graph the equation \(6x - y = 36\), plot the x and y intercepts on the graph first. The x-intercept is at (6,0) so mark this point on the x-axis. The y-intercept is at (0,-36) so mark this point on the y-axis. Draw a line through these two points. This represents the line solution of the given equation.

Key concepts

X-Intercept

The x-intercept of a graph is the point where the line crosses the x-axis. This is also the value of the variable 'x' when 'y' is set to zero. To find this point, in our example with the equation 6x - y = 36, we will make 'y' equal to zero and solve for 'x'. By substituting 'y' with zero, the equation becomes 6x = 36. Dividing both sides by 6 gives us x = 6. Therefore, the x-intercept for this equation is at the point (6, 0). When plotting this on a graph, you locate the number 6 on the x-axis and place a point right there.

Remember that the x-intercept provides valuable information about the graph, as it shows where the line will touch the x-axis as it stretches horizontally. This is a key part of graphing linear equations and understanding its behaviour along the x-axis.

Y-Intercept

In contrast, the y-intercept is where the graphed line crosses the y-axis. It is the value of 'y' while 'x' is zero. To find it in the given equation 6x - y = 36, you set 'x' to zero. That simplifies our equation to -y = 36. To solve for 'y', we multiply both sides of the equation by -1 yielding y = -36. Thus, the y-intercept is at (0, -36).

On a graph, you locate -36 on the y-axis and mark your point there. It is critical to note the y-intercept because it tells us where the line will begin if it was plotted from the bottom to top, or vice versa. This point is fundamental in sketching the trajectory of the line on the graph.

Solve for Y

Being able to solve for 'y' is a vital aspect when dealing with linear equations. It's often the first step in graphing the equation since it helps to rearrange the equation into the slope-intercept form y = mx + b, where 'm' is the slope and 'b' is the y-intercept. Let's take our equation 6x - y = 36 and solve for 'y'. We isolate 'y' by moving it to one side of the equation, resulting in y = 6x - 36.

In this form, it is much clearer to identify the slope (6) and y-intercept (-36). With this information alone, one can graph the equation with ease by first plotting the y-intercept, and then using the slope to find another point on the line.

Plotting Points

After finding the x-intercept and y-intercept, plotting points is the next step to visualize the equation on a graph. To plot these points correctly, you start with the x-intercept and y-intercept, which are (6, 0) and (0, -36) respectively for our equation. On graph paper, find the point where 'x' is 6 and 'y' equals 0, this is the x-intercept. Then, locate the point where 'x' is 0 and 'y' is -36, this is the y-intercept.

Once both intercepts are plotted, draw a straight line through them. This line represents all the possible solutions to the equation 6x - y = 36. By plotting points and connecting them, you are turning an algebraic expression into a visual representation that can be analyzed and understood more intuitively.