So we’ll start with slope. Slope is the measure of the steepness of a line. Slope is also known as the constant rate of change. So another way we can think about slope is our change. In y, divided by our change in x, or also rise over run okay, so what is rate of change for each of the lines shown? So what we’re going to do is we’re going to take two of our points. It doesn’t matter which ones and we’re going to make a right triangle. So, if i’m, looking at this first example here, i’ve got this coordinate here and this coordinate here. Okay, i make a right triangle with those two coordinates. My rise is i’m going down one. Two three, so my rise is negative. Three i run one two, so my run is two. If we refer back to earlier, slope is rise over run, so it’s negative three over two for my slope. Now you can pick any two points on that graph and you’ll. Always get negative three over two so say i picked the first one in the last one. I went down one two, three, four, five, six, seven, eight and nine. So my rise was negative. Nine and my run was one two three four five, six, so positive. Six negative nine over six. I can divide both of these by three and that would be the same thing as negative three over two. So, no matter what two points you pick on a graph just as long as there are two points that cross through the corner of a square, because we know their exact value, you will always get the same thing for your slope.
So if i look at the second example, okay, i pick any two points as long as they’re crossing through the corner of a square. So my rise here is one two, three, four, five, six, seven, eight nine and ten. So my rise is ten. My run is one two three four, so my run is four, so i get the fraction 10 over four, which both can be divided by two. So i can reduce that fraction to five over two. So my slope, for that example, would be five over two or five halves. Okay. The next example asks us for a couple different things: it wants to know the slope, the y intercept and the x intercept of the graph below. So to find the slope, we need to pick two points on our line where it crosses through the corner of a square so i’m crossing through the corner of a square here and here. So i take those two points and i make a right triangle. So one two and three so my rise was negative. Three one, two three and four, so my run was four. So my slope is my rise or negative. Three divided by my run, which is four so my slope is negative. Three fourths my x intercept is where my graph crosses my x axis. So this point right here is where my graph crosses the x axis, so that coordinate would be negative. Four comma zero, so my x intercept is negative: four comma zero.
My y intercept is where my graph crosses my y axis, so this coordinate here is where my graph is crossing my y axis and that coordinate would be 0 negative 3.. So my y intercept is 0 negative. 3.. If i look at the last example on this page, it wants to know if the slope of the line is multiplied by negative 1 and the y intercept is decreased by two which linear equation represents these changes. Okay, so first we’ve got to understand: slope intercept form, so slope intercept form is y equals mx plus b. Where m is our slope and b? Is our y intercept? So i need to find those two things to start off my equation. So for my slope, i pick two points or i cross through the corner of a square, so i can pick here and here so i make a right triangle. My rise is 1 2. My run is 1., so my slope would be 2 over 1, which is the same thing as just 2.. My y intercept is where i cross. The y axis right here is where my graph is crossing the y axis that coordinate there is zero, comma positive. Three. So because it’s, a positive three, my b or y intercept is plus 3., so my equation would be y equals 2x plus 3.. So in the problem it says multiply, the slope is multiplied by negative one, so that means i’m multiplying two by negative one, and then my y intercept is decreased by two units, so i’m subtracting two from my y intercept.
So now i’d have y equals negative 2x because 2 times negative 1 would be negative, 2 and then positive 3 minus 2 would be positive, 1. and then my equation there would be y equals negative, two x, plus one which graph represents the equation. Two x e minus three y equals six, so i need to convert this to slope, intercept form, which means i need to solve for my y. I need to get that y all by itself. Everything else has to go so the first thing that i’m going to deal with is that 2x it’s a positive 2x. So that would be like addition, so the opposite would be subtraction. So i subtract 2x on both sides of my equation. On the left side. They cancel out and i’m left with negative 3y on the right side. I have negative 2x plus 6.. Then i need to again get my y by itself. So i’ve got to get rid of that negative 3. Well, the negative 3 is right next to the y. So it means i’m multiplying those two terms together. The opposite of multiplication is division so i’m, going to divide everything in my problem by negative three on the left side. They cancel each other out, and all i have left over is just simply y negative. Two divided by negative three would give me a positive two thirds and then x, positive 6, divided by negative 3, would give me a negative 2.
, so i’m. Looking for an equation that has the slope of 2 3 and a y intercept of negative 2. well, i can get rid of f and g because they both have a y intercept of positive 2, not negative 2., and then i know my answer is not j, Because my graph, if i follow it from left to right, is going down, which means i have a negative slope and if we look at our problem, our slope is positive. So we know it has to be h. The next one says how the graph of y equals 2x minus 5 compare to the graph of y equals 3x minus 5.. So, under normal circumstances, i would just simply put that into my calculator and see how they differ from each other right now, i’m going to go ahead and graph, each of them by hand. So the first one is y equals 2x minus five, so i’m, going to do that, one in orange, so minus five means my y intercept is negative five and then my slope is two which is the same thing as two over one. So i go up two over one, so up two over one that’s. My next point and i can keep graphing my points by going up two and over one. So then my graph would look something like that, and then i will do my other equation and let’s go red so again at minus five. So my y intercept is at negative five, but my slope is three: so it’s three over one, so up three over one and that’s.
My next point up three over one up three over one and i graph each of my coordinates and so it’ll look something like this. So i need to figure out which equation or which statement is true. So the slope of y equals 2x minus 5 is less steep. Okay, so that’s, potentially true the slope of 2x minus 5 is steeper. That is incorrect. The graph of y equals 2x minus 5 has a greater y intercept that’s, not true. They’Re the same. The graph of y equals 2x minus 5 has a smaller y intercept that’s, not true again, they’re the same so my answer: choice there would have been answer: choice, f, the slope of y equals 2x. Minus 5 is less steep. The next one says which equation describes. The line that passes through the point 4 7 and is parallel to the line represented by the equation: negative 3x plus y equals 4.. So the easiest way to go about this one is to solve this equation for y and then graph it graph your answer, choices and see which ones are parallel, the ones that are parallel just look at the table and see if it has the coordinate 4. Comma. 7., that would be the easiest way to do it, since i don’t have a calculator right now, i’m going to go ahead and do it by hand, so i need to solve for y, so i’ll add 3x on both sides, so i can get the y all By itself i add 3x because it’s negative, which is the same thing as subtraction.
The opposite would be addition, so i get y equals 3x plus 4.. In parallel lines. Our slope is always the same, so i know i’m going to have a slope of 3.. So all i need to do is plug in this coordinate and then solve for my b value. So for my new equation, it has to be y equals 3x plus b, and then i plug in my x and y value from that coordinate to find my b value. So my y value is 7. My x value is 4 and then i’m solving for b 3 times 4 is 12., so 7 equals 12 plus b and in order to get my b by itself, i just need to subtract 12 on both sides and i get five equals b. So my equation is y, equals three x plus five or i’m. Sorry, that should be a negative five that’s, a negative five which would make this minus five. So then our answer choice would be g all right. Next, one says which line has a slope of zero. So zero slope means our line is going straight across okay, so that would be line w, which line has undefined slope.