So, welcome back. Now you've completed some more work on Grant's glasses. You looked at simplification of equations, you looked at graphing, and finding equations of lines. And you also looked at calculating the midpoint and finding distances between points. Let's review some problems.
Let's begin by reviewing solving equations. Here's a problem for you to try. Let's solve for x.
There are variety of ways that we could start this problem. But we're going to start by multiplying both sides of the equation by 4 and then distributing the 3 with the quantity x plus 2. You can continue simplifying the problem combining like terms. This gives us negative 62 equals 9x. You can now divide both sides by 9, and our solution is negative 62 divided by 9 equals x.
Here's an expression for you to simplify.
There are a variety of ways that we could start this problem. I'll start by simplifying what's under the radical sign, and distributing the 2x with the quantity x plus 1. So if you look at what's under the radical sign, the square root of 2x cubed, broken out, that's the square root of 2 times x squared times x. x squared comes out as x times the square root of 2x. And next, we're going to distribute the 2x with the quantity x plus 1. This gives us x times the square root of 2x, raised to the fourth power, plus 2x squared, plus 2x. Now, let's simplify what's in the parentheses. This gives us 4x to the 6th, plus 2x squared, plus 2x.
I've drawn a few lines for you to look at. >> Please type the letter of the graph that represents each equation.
Take a few moments to look at the solutions. Also note that there are two types of equations, vertical and horizontal. represented amongst these four equations.
Now please tell me the slope and y-intercept for each equation.
Please take a couple of minutes to look at the solutions. You might notice there are two special types of lines we've talked about before. The first one being a horizontal line, which has a slope of 0, And the second one a vertical line which has an undefined slope. And note that it has no y-intercept.
Here are two points, u and c. Please find the distance between these two points.
Recall the formula for finding the distance between two points. Substituting in the formula for u and c, we get. And when we simplify, we obtain the square root of 25 plus 9, which gives us the square root of 34.
Great. I bet you thought you are done with this problem, but not quite yet. Let's keep going. We want you to first, find the midpoint between u and c, so let's try that now.
Recall how we determine the midpoint between two points. Our x coordinate is found by adding the two x coordinates and diving by two, and the y coordinate is found by adding the two y coordinates and dividing by two. Once we plug in our points, we obtain a midpoint of three halves and nine halves.
Find the equation of a line that is perpendicular to the line uc and goes through the midpoint between u and c. In order to complete this question, we need to brainstorm the process to do this. So, first of all, you're going to want to find the slope of line uc. Then, you're going to want to find the perpendicular slope to uc. And finally, use the slope intercept form, y equals mx plus b, to find the equation of the line.
Let's take a couple [of] minutes to look at the solutions. When we're looking at the slope of line 'uc,' you can see that our slope works out to be 3/5. Finding the perpendicular slope is then relatively easy. We find the inverse of the slope, which is negative 5/3. Now it's time to find the equation of this line. Let's first recall our midpoint, 3/2 and 9/2, and our perpendicular slope, which is negative 5/3. Please insert those values into the point-slope form and solve. And our equation in slope intercept form is y = -5/3 x + 7.