The reason why will be obvious in the next section. The other lines now have negative slopes and slant downwards from left to right. Each slope is the negative for the same-color line in Graph A. Jump to: navigation , search. In the graph above, we need to go up 2 spaces from the first point to be in line with the second point. This becomes our numerator. Then we count how far we need to "run" horizontally to get from the first point to the second point.
The second point is 8 units to the right of the first point. This becomes our denominator. We simplify the fraction and get a slope of. We pick two points on the line and then we subtract the y -values to get the numerator. Then subtract the x -values and make this the denominator.
The formula is super handy if the points are decimals or fractions. The important thing to remember is to keep the points in the same order in the numerator and denominator. So , which is the same answer we got using the other method. Using the slope formula,.
You do not need the graph to find the slope. You can just use the coordinates, keeping careful track of which is Point 1 and which is Point 2. Notice that regardless of which ordered pair is named Point 1 and which is named Point 2, the slope is still 3. What is the slope of the line that contains the points [latex] 3, No matter which two points you choose on the line, they will always have the same y -coordinate.
But there are two other kinds of lines, horizontal and vertical. What is the slope of a flat line or level ground? Of a wall or a vertical line? You can also use the slope formula with two points on this horizontal line to calculate the slope of this horizontal line. So, when you apply the slope formula, the numerator will always be 0. Zero divided by any non-zero number is 0, so the slope of any horizontal line is always 0.
How about vertical lines? In their case, no matter which two points you choose, they will always have the same x -coordinate. So, what happens when you use the slope formula with two points on this vertical line to calculate the slope? But division by zero has no meaning for the set of real numbers. Because of this fact, it is said that the slope of this vertical line is undefined. This is true for all vertical lines—they all have a slope that is undefined.
When you graph two or more linear equations in a coordinate plane, they generally cross at a point.
However, when two lines in a coordinate plane never cross, they are called parallel lines. You will also look at the case where two lines in a coordinate plane cross at a right angle. These are called perpendicular lines. The slopes of the graphs in each of these cases have a special relationship to each other. Parallel lines are two or more lines in a plane that never intersect.
Examples of parallel lines are all around us, such as the opposite sides of a rectangular picture frame and the shelves of a bookcase. Perpendicular lines are two or more lines that intersect at a degree angle, like the two lines drawn on this graph. These degree angles are also known as right angles. Perpendicular lines are also everywhere, not just on graph paper but also in the world around us, from the crossing pattern of roads at an intersection to the colored lines of a plaid shirt.
The slope of both lines is 6. They are not the same line. The slopes of the lines are the same and they have different y -intercepts, so they are not the same line and they are parallel. Two non-vertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other.
As you move from the point -1, -5 to the point 2, 10 , the line has a rise of 15 and a run of 3, so the slope of the line is. The m is the slope of the line. And b is the b in the point that is the y -intercept 0, b. Solve for y. The slope is , and the y -intercept is 0,.
You can also find the slope of a straight line without its graph if you know the coordinates of any two points on that line. Every point has a set of coordinates: an x -value and a y -value, written as an ordered pair x , y. The x value tells you where a point is horizontally. The y value tells you where the point is vertically. Consider two points on a line—Point 1 and Point 2.
Point 1 has coordinates x 1 , y 1 and Point 2 has coordinates x 2 , y 2. The rise is the vertical distance between the two points, which is the difference between their y -coordinates. So, or. So you are going to move from Point 1 to Point 2. A triangle is drawn in above the line to help illustrate the rise and run. You can see from the graph that the rise going from Point 1 to Point 2 is 4, because you are moving 4 units in a positive direction up. Using the slope formula,.
You do not need the graph to find the slope. You can just use the coordinates, keeping careful track of which is Point 1 and which is Point 2. Ordered Pair. Point 1. Point 2. In that case, putting the coordinates into the slope formula produces the equation.
What is the slope of the line that contains the points 5, 5 and 4, 2? Substitute the values into the slope formula and simplify. The slope is 3. The example below shows the solution when you reverse the order of the points, calling 5, 5 Point 1 and 4, 2 Point 2. Notice that regardless of which ordered pair is named Point 1 and which is named Point 2, the slope is still 3.
What is the slope of the line that contains the points 3, The slope is The correct answer is. Put the coordinates into the slope formula consistently:. You have interchanged the rise and the run. Advanced Question. What is the slope of a line that includes the points and? It looks like you inverted the rise and the run.
Use the formula to find the slope. It looks like you subtracted either the y or x coordinates in the wrong order. Make sure you subtract , then , and then calculate the slope. Using the formula for slope, , you found that.
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