The Graph of a Linear Equation in Two Variables Is a Line 05:34 minutes

Video Transcript

Transcript The Graph of a Linear Equation in Two Variables Is a Line

Meet Evelyn K. Neevil. For her famous grandfather's birthday, she's planning a death-defying jump. But, unlike her grandfather she prefers not to break any bones while doing her stunts. To avoid this fate, she and her team will investigate Graphing Linear Equations with Two Variables.

Evelyn knows the equations for the two ramps she plans to use for her jump. The equation of the first ramp in slope-intercept form is 'y' equals one-half 'x' plus 1 and the second equation, also in slope-intercept form, is 'y' equals negative one-fourth 'x' plus 5. But, what do these ramps look like? We know the formula for each ramp is a linear equation as all variables in both equations are raised to the first power we know that the graphs of such linear equations are lines.

Let's help Evelyn figure out how steep each ramp is. To do this, we need to look at the slope of each line. Let's graph the two linear equations. From the graph, we'll be able to see how deadly Evelyn's jump will be. How do we graph a linear equation? An easy way is to calculate the 'x-' and y-intercepts. Let's first find the y-intercept. The y-intercept is the point where the graph of the line intersects the y-axis. The x-coordinate of the y-intercept is always zero. So, to find the y-coordinate of the y-intercept just set 'x' equal to zero. We can plug zero into the equation and then solve for 'y'. Since anything times zero is equal to zero, we’re left with 'y' equals 10. So, our first coordinate is the ordered pair, (0, 10). Remember, coordinates are ordered pairs that indicate specific positions on the coordinate plane. See how this coordinate sits on the y-axis? But how do we find the coordinate, or point, that intersects the x-axis? That’s right! We just have to set ‘y’ equal to zero and solve for ‘x’! We get zero equals one-half 'x' plus 10. Using opposite operations, we first move the 10 to the other side. And since ‘x’ is multiplied by 1/2 to isolate the variable we need to multiply both sides by the inverse of 1/2, which is 2 giving us -20 = ‘x’. Thus, the x-intercept is the ordered pair, (-20, 0).

Now that we know the two intercepts we can draw a line connecting them. Now we can see how steep the first ramp is! With a slope of 1 over 2. Looks pretty steep, right?! Now, let’s find the 'x-' and y-intercepts for the second linear equation. If we recognize that the equation is in slope-intercept form we know the y-int will simply be 5. So, since our 'x' coordinate is zero, our 'y' intercept is (0, 5). Just like before. To find the x-intercept we set ‘y’ equal to zero and solve for ‘x’. We get zero equals negative one-fourth 'x' plus 5. Using opposite operations, we first move the 5 to the other side by subtracting and since we multiplied ‘x’ by -1/4, we need to multiply both sides by -4 to isolate the 'x' giving us ‘x’ = 20. So, the line 'y' equals negative one-fourth 'x' plus 5 intersects the x-axis at (20, 0) and intersects the y-axis at (0, 5). Connecting the dots, we can easily see how steep the second ramp is with a slope of negative one over four.

Looks like all the preparations are ready, and Evelyn K. Neevil is ready for her jump. There she goes!!! Will...she...make iiiiit!?!?