When you think about geometry, the first thing that comes to mind is probably the classic line equation: y = mx + b. This equation describes a line segment, or the length of a path that connects two points.
The y variable in this equation represents the height of the line (where the point is located), m represents the slope of the line (how steep it is), x represents the starting point of the line, and b represents the ending point of the line.
When given two points on a line, you can find all sorts of useful information about that line using this equation. For example, you can find what the slope of the line is, what its midpoint is, and so on.
This article is going to go into more detail about some less-known lines and how to identify them using only your basic knowledge of algebra.
y-intercept formula
The last slope equation type is the y-intercept formula. This refers to the point at which a line intercepts the y-axis, or the value where the line would intersect the y-axis if it were extended all the way down.
To find the y-intercept of a line, you must find two of its points and use a simple formula. The hardest part is finding two of its points, as you have to do this in reverse.
The y-intercept formula is b=y−x, where b is the y-intercept, y is the higher point on the line, and x is the lower point on the line. This makes sense when you think about extending the line down to where it intercepts the y-axis.
Point-slope formula
The point-slope formula for the equation of a line is given by
Y=mx+b, where m is the slope of the line and (x,y) is a point on the line. b is the y-coordinate of the point on the line, and m is the difference in m and b.
The points (x,y) can be either numbers or coordinates. For example, (4,8) or (4,-2) are points on the line y=4x-2.
How to find slope: The slope can be found by taking two points on the line and doing some math.
Linear equation of a line
Now let’s look at how to find the equation of a line that passes through a given point and has a given slope.
The equation of a line can be written as y = mx + b, where m is the slope and b is the y-intercept. The x-intercept is found by putting 0 for y, which would make the equation y = x + b, and -b for x, which would make the equation b = -x.
To find the slope and y-intercept of a line, you need to find two points on the line and use the points to find the formula for the line.
Slope
The slope of a line is represented by the letter m. The slope of a line describes how the line moves up or down as it moves to the right or left.
There are two ways to describe the slope of a line. The first way is to describe the change in y-values relative to the change in x-values. For example, if a line changed 4 y-values for every 1 x-value change, then we would say that the slope is 4.
The second way is to use the letter m, which stands for the ratio of y-offset to x-offset. If there is a 1 unit difference in the y-values and a 1 unit difference in the x-values, then m=1.
The equation n=a·b represents a linear equation where n is defined as an unknown variable that responds according to a, b, and all possible combinations of a and b.
Y-intercept
The y-intercept of a linear equation is the value where the line crosses the y-axis. The y-intercept is also called the f(x) value, where f(x) is the variable in the equation for the line.
For example, if the equation of a line is y=2x−3, then 2 is the y-intercept because when x=0, then y=2. The line crosses the y-axis at 2.
The slope of a linear equation can be represented in many different forms, including a ratio, a quotient, and an algebraic expression. By looking at these different representations, you can identify what makes up slope of a linear equation.
All of these representations have one thing in common: they all have an implicit multiplication symbol between the x- and y-values.
Point-slape formula
The point-slope formula describes the equation of a line that passes through a given point and has a given slope. The line is defined by the following equation:
y − y 1 = m(x − x 1 )
where y is the vertical coordinate or the line’s dependent variable, x is the horizontal coordinate or the line’s independent variable, m is the slope, and (x 1 , y 1 ) is the given point.
The first step in finding this equation is to find the slope, m. The slope can be found by finding the difference between the coordinates and dividing that by the coordinate of one point. For example, if we have (3, 4) as one point and (5, 6) as another point on the line then: mx=6−5=1m=1/5m.
When finding which points are on a line, it does not matter which coordinate is put first. Therefore, (3, 4) and (4, 3) are still on the same line.
The second step in finding this equation is to find what value to put for y in order for x to be equal to its corresponding x 1 . Then y−y 1 must be equal to m(x−x 1 ) so that x equals y. This means that y must be equal to mx−yx 1 .
If we take our original example of (4,-1) being on a line with slope 2 then (-2,-1)=2(-4,-1)=2 so (-2,-1)=(-2,-1))mustbeonthisthesameline.
Thus we have found our answer! Any other points with (-2,-1) would also be on this same line.
Which Equation Represents a Line That Passes Through (Intersection) (-4,) And Has A Slope Of ? – Wikipedia article about intersection geometries <iframe src="//cdn.toubiiro.com/player/clip_player_old.js?i=ev9f6d7p6&pbid=eb8229efb&pcode=ev9f6d7p6&iurl=http%3A%2F%2Fwww.toubiiro-tvchannel-tvchannel24hrs-net%2Fev9f6d7p6%3Fiurl%3Dhttps%253A%252F%252Fwww.-google.-com%252Famphtml%253Fuurl%253Dhttps://en.-wikipedia.-org/-intersectiongeometry-intersectio%20n%20geometry%20amp;;title=-Intersection%20Geometry;rel=canonical;;link=https://en.-wikipedia.
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Now that you understand the basics of linear equations, you can use them in more real-life situations. You can use them to solve real-world problems like finding the distance between two points or how much a quantity will increase or decrease depending on circumstances.
For example, let’s say a person travels 4 miles east from a starting point and then travels 7 miles north to another point. How far is the second point from the starting point?
To solve this problem, you must first write an equation that represents the line that passes through both points. To do this, take the x-coordinate of the second point and subtract the x-coordinate of the starting point. Then, take the y-coordinate of the second point and subtract the y-coordinate of the starting point. Put these numbers into an equation and solve for x or y to get your answer.
Summary
The equation of a line can be determined by finding the average of the coordinates of both endpoints, then putting that average over the slope.
The average of the x-coordinates is 4, so your equation would be Y = 4. The average of the y-coordinates is , so your equation would be Y = .
Now that you have these two equations, you can put them together and solve for X or Y to find the slope. Doing this reveals that the line passes through (4, ) and has a slope of .
This process can be difficult to do mentally, so writing it down and solving for X or Y can help clarify what is happening. It is also possible to solve for X and Y separately, which could reveal more information about the line.
