How to Calculate the slope, slope and origin of a line

The slope of a line measures its slope. ^{[1] X Research Source} You can also say that it is the vertical change (rise) on the horizontal change (run) or the rise of the line vertically relative to its movement horizontally. Finding the slope of a line or using it to find points on a line are important skills in economics, ^{[2] X Geoscience Research Resources} , ^{[3] X Research Resources study} accounting/finance and many other fields.

Table of Contents

Steps

Get familiar with basic shapes:

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Find the slope using the graph

Select two points on the line. Represent and record their coordinates on a graph.

Remember, the coordinate is first and the coordinate is behind.
For example, you can choose scores (-3, -2) and (5, 4).

Determines the vertical change between two points. To do this, you have to compare the two-point coordinate difference. Start with the first point, which is far to the left of the graph, and move until it meets the coordinates of the second point.

Vertical shifts can be positive or negative, meaning you can shift up or down. ^{[4] X Research Source} If our line moves up and to the right, the change in coordinates will be positive. If the line moves down and to the right, the vertical change is negative. ^{[5] X Research Sources}
For example, if the coordinate of the first point is (-2) and the second point is (-4), you will add 6 points or your vertical change is 6.

Determine the horizontal change between two points. To do this, you have to compare the difference between the two points’ coordinates. Start with the first point, which is the farthest left of the graph, and progress until you get the coordinates of the second point.

The horizontal shift is always positive, meaning you can only go from left to right and never vice versa. ^{[6] X Research Source}
For example, if the coordinate of the first point is (-3) and the second point is (5), you will have to add 8, which means your horizontal shift is 8.

Calculate the ratio of horizontal change to vertical change to determine the slope. The slope is usually in the form of a fraction, but sometimes it is an integer.

For example, if the vertical shift is 6 and the horizontal shift is 8 then your slope is $6$ $8$ ${displaystyle {frac {6}{8}}}$ ${frac {6}{8}}$ . Simplify we get: $3$ $4$ ${displaystyle {frac {3}{4}}}$ ${frac {3}{4}}$ .

Find the slope equal to two given points

Formula setting $m$ $=$ $y$ $2$ $-$ $y$ $first$ $x$ $2$ $-$ $x$ $first$ ${displaystyle m={frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$ $m={frac {y_{{2}}-y_{{1}}}{x_{{2}}-x_{{1}}}}$ . Where, m = slope,

(

x

first

,

y

first

)

{displaystyle (x_{1},y_{1})}

(x_{{1}},y_{{1}})

= coordinates of the first point,

(

x

2

,

y

2

)

{displaystyle (x_{2},y_{2})}

(x_{{2}},y_{{2}})

= coordinates of the second point.

Remember that the slope is equal to the vertical change over the horizontal change, or $r$ $i$ $S$ $e$ $r$ $u$ $n$ ${displaystyle {frac {rise}{run}}}$ ${frac {rise}{run}}$ . You are using a formula to calculate the (vertical) coordinate change over the (horizontal) coordinate change. ^{[7] X Research Sources}

Substitute coordinates into the formula. Make sure you have substituted the coordinates of the first point (

(

x

first

,

y

first

)

{displaystyle (x_{1},y_{1})}

(x_{{1}},y_{{1}})

) and the second point (

(

x

2

,

y

2

)

{displaystyle (x_{2},y_{2})}

(x_{{2}},y_{{2}})

) in the correct place in the formula. Otherwise, the obtained slope will be incorrect.

For example, with two points (-3, -2) and (5, 4), your formula would be: $square meter$ $=$ $4$ $-$ $($ $-$ $2$ $)$ $5$ $-$ $($ $-$ $3$ $)$ ${displaystyle m={frac {4-(-2)}{5-(-3)}}}$ $m={frac {4-(-2)}{5-(-3)}}$ .

Do the math and reduce if possible. You will get the slope as a fraction or an integer.

For example, if your slope is $square meter$ $=$ $4$ $-$ $($ $-$ $2$ $)$ $5$ $-$ $($ $-$ $3$ $)$ ${displaystyle m={frac {4-(-2)}{5-(-3)}}}$ $m={frac {4-(-2)}{5-(-3)}}$ , you should put $4$ $-$ $($ $-$ $2$ $)$ $=$ $6$ ${displaystyle 4-(-2)=6}$ $4-(-2)=6$ in the denominator (Remember that when subtracting negative numbers, you add) and $5$ $-$ $($ $-$ $3$ $)$ $=$ $8$ ${displaystyle 5-(-3)=8}$ $5-(-3)=8$ in the numerator. You can shorten $6$ $8$ ${displaystyle {frac {6}{8}}}$ ${frac {6}{8}}$ wall $3$ $4$ ${displaystyle {frac {3}{4}}}$ ${frac {3}{4}}$ and thus: $m$ $=$ $3$ $4$ ${displaystyle m={frac {3}{4}}}$ $m={frac {3}{4}}$ .

Find the origin when the slope and a point . are known

Formula setting $y$ $=$ $m$ $x$ $+$ $b$ ${displaystyle y=mx+b}$ $y=mx+b$ . Where y = coordinate of any point on the line, m = slope, x = coordinate of any point on the line, and b = origin.

$y$ $=$ $m$ $x$ $+$ $b$ ${displaystyle y=mx+b}$ $y=mx+b$ is the equation of a straight line. ^{[8] X Research Sources}
The origin coordinate is the point at which the line intersects the vertical axis.

Substitute the slope and coordinate values of a point on the line. Remember, the slope is equal to the vertical change over the horizontal change. If you need to find the slope, refer to the instructions above.

For example, if the slope is $3$ $4$ ${displaystyle {frac {3}{4}}}$ ${frac {3}{4}}$ and (5,4) is a point on the line, so the obtained formula is: $4$ $=$ $3$ $4$ $($ $5$ $)$ $+$ $b$ ${displaystyle 4={frac {3}{4}}(5)+b}$ $4={frac {3}{4}}(5)+b$ .

Complete and solve the equation, find b. First, multiply the slope and the coordinates. Subtracting both sides of this product, we get b.

In the example problem, the equation becomes: $4$ $=$ $3$ $3$ $4$ $+$ $b$ ${displaystyle 4=3{frac {3}{4}}+b}$ $4=3{frac {3}{4}}+b$ . Subtract two sides for $3$ $3$ $4$ ${displaystyle 3{frac {3}{4}}}$ $3{frac {3}{4}}$ , we get $first$ $4$ $=$ $b$ ${displaystyle {frac {1}{4}}=b}$ ${frac {1}{4}}=b$ . So, the origin is $first$ $4$ ${displaystyle {frac {1}{4}}}$ ${frac {1}{4}}$ .

Check calculation. On the coordinate plot, represent a known point and then, based on the slope, draw a line through that point. To find the angle, find the point at which this line crosses the vertical axis.

For example, if the slope is $3$ $4$ ${displaystyle {frac {3}{4}}}$ ${frac {3}{4}}$ and given point is (5,4), take a point at coordinate (5,4) and draw other points along the line by counting left 3 and down 4. When drawing a line passing through the points, the line is drawn obtained should intersect the vertical axis at the point located on the origin (0,0).

Find the origin when the slope and the origin are known

Formula setting $y$ $=$ $m$ $x$ $+$ $b$ ${displaystyle y=mx+b}$ $y=mx+b$ . Where: y = coordinate of any point on the line, m = slope, x = coordinate of any point on the line, and b = origin.

$y$ $=$ $m$ $x$ $+$ $b$ ${displaystyle y=mx+b}$ $y=mx+b$ is the equation of a straight line. ^{[9] X Research Source}
The horizontal origin is the point at which the line passes through the horizontal axis.

Generation of angle numbers and tossing the origin into the formula. Remember, the slope is equal to the vertical change over the horizontal change. If you need assistance in finding the slope, you can refer to the instructions above.

For example, if the slope is $3$ $4$ ${displaystyle {frac {3}{4}}}$ ${frac {3}{4}}$ and the origin is $first$ $4$ ${displaystyle {frac {1}{4}}}$ ${frac {1}{4}}$ , the obtained formula will be: $y$ $=$ $3$ $4$ $x$ $+$ $first$ $4$ ${displaystyle y={frac {3}{4}}x+{frac {1}{4}}}$ $y={frac {3}{4}}x+{frac {1}{4}}$ .

Let y be 0. ^{[10] X Research Source} You are looking for the origin, the point at which the line intersects the horizontal axis. At this point, the coordinate will be 0. So, if y equal to 0 and solve the obtained equation to find the corresponding coordinate, we get the point (x, 0) – which is the original coordinate to find.

In the example problem, the equation becomes: $0$ $=$ $3$ $4$ $x$ $+$ $first$ $4$ ${displaystyle 0={frac {3}{4}}x+{frac {1}{4}}}$ $0={frac {3}{4}}x+{frac {1}{4}}$ .

Complete and solve the equation, find x. First, subtract both sides of the origin. Next, divide both sides by the slope.

In the example problem, the equation becomes: $-$ $first$ $4$ $=$ $3$ $4$ $x$ ${displaystyle {frac {-1}{4}}={frac {3}{4}}x}$ ${frac {-1}{4}}={frac {3}{4}}x$ . Divide both sides by $3$ $4$ ${displaystyle {frac {3}{4}}}$ ${frac {3}{4}}$ , get: $-$ $4$ $twelfth$ $=$ $x$ ${displaystyle {frac {-4}{12}}=x}$ ${frac {-4}{12}}=x$ . Simplified we have: $-$ $first$ $3$ $=$ $x$ ${displaystyle {frac {-1}{3}}=x}$ ${frac {-1}{3}}=x$ . So the point at which the line passes through the horizontal axis is $($ $-$ $first$ $3$ $,$ $0$ $)$ ${displaystyle ({frac {-1}{3}},0)}$ $({frac {-1}{3}},0)$ . So the origin is $-$ $first$ $3$ ${displaystyle {frac {-1}{3}}}$ ${frac {-1}{3}}$ .

Check calculation. On the coordinate plot, plot your origin, then, based on the slope, draw a line. To find the origin, find the point at which the line intersects the horizontal axis.

For example, if the slope is $3$ $4$ ${displaystyle {frac {3}{4}}}$ ${frac {3}{4}}$ and the origin is $($ $0$ $,$ $first$ $4$ $)$ ${displaystyle (0,{frac {1}{4}})}$ $(0,{frac {1}{4}})$ , point representation $($ $0$ $,$ $first$ $4$ $)$ ${displaystyle (0,{frac {1}{4}})}$ $(0,{frac {1}{4}})$ and draw other points along the line by counting to the left 3 and down 4 and then to the right 3 and up 4. When drawing a line through the points, the resulting line should intersect the horizontal axis only slightly to the left. origin (0,0).

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Steps

Find the slope using the graph

Find the slope equal to two given points

Find the origin when the slope and a point . are known

Find the origin when the slope and the origin are known

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