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In the previous sections, we've discussed individual lines and their behavior. Now that we understand the basics of how a line behaves by itself, we're ready to use lines as a means of understanding more complex concepts, specifically how lines can behave together.
The graph below shows two lines on the coordinate plane. The slope and intercepts of these lines can be manipulated by dragging the sliders to the left
The graph below shows two lines on the coordinate plane. These lines can be manipulated by dragging the sliders to the left. Try playing around with the sliders and noticing how the lines interact.
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Discussion
What happens when
m1=m2 and c1=c2?
m1=m2 and c1=c2?
m1=m2?
m1=−m21?
For the two lines, look at their “tilt” (slope) and how they sit on the page, along with their intercepts:
When m1=m2, both lines rise at the same steepness.
If c1=c2, they run side by side, always the same gap apart and never meet.
If c1=c2, they lie exactly on top of each other—every point matches, so you see just one line.
When m1=−m21, their tilts make them cross like a “plus sign,” meeting at a perfect right-angle corner.
When the tilts differ in any other way (m1=m2 and m1=−m21), they swing past each other in an X-shape, crossing once at a slanted angle.
There are three different kinds of "relationships" that two lines can have to each other.
1) The lines never touch
If two lines never touch, this means that no matter how big or how small the x we input is, there can be no point where plugging in the same x value returns the same y value. This does not necessarily happen when one line has an intercept at y=1,000,000,000 and another at y=−1,000,000,000, because we have infinite x values to work with. It doesn't matter if the point where they touch is a number too big for us to understand, because we still know that they will touch.
The determining factor in whether or not lines will ever touch is the gradient. If two lines have the same gradient and different intercepts, then the distance between the lines at x=0,
will be the distance between the two lines at every value of x. Remember that the gradient of a line m tells you to take mx vertical steps for every x horizontal steps. When two lines have the exact same gradient, moving x horizontal steps on both lanes will change both of their y values by the same amount, so they will stay the exact same distance apart everywhere.
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When two lines never touch, we say that they are parallel.
2) The lines always touch
In other words, they are the same line. This means that they are defined the exact same way, so even if they look different at a first glance (written in different forms, use different fractions, or something else), the relationship between x and y given by their equations is the exact same in both cases.
This is very similar to the first case, except this time, both the slope and the intercepts must be the same.
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Parallel lines
Two lines are parallel when they have the same gradient m and they do not intersect:
In this case, the system of equations formed by the two lines has no solutions.
If the lines have the same gradient and they intersect, then they must be the same line.
Checkpoint
Two lines are plotted on the coordinate plane.
The first line is defined by the gradient-intercept form equation y=−3x+5. The second line is defined by the standard form equation 3x+y−8=0.
Explain whether the two lines are parallel.
Select the correct option
3) The lines touch exactly once
This is the easiest condition to fulfill. Two lines will touch exactly once as long as their slopes are not the same. Since we're dealing with straight lines, they cannot touch more than once.
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To find the x coordinate of the point where the two lines intersect, set the equations equal to each other. Solving for x will tell you the x coordinate where the equations take on the same y value. Substituting this x value back into either equation will give you the y value of the point where they intersect.
In the example above,
Hence the two lines intersect at the point (−37,−38).
Intersections of straight lines
Suppose we have the straight lines y=3x−2 and y=2−3x. Where do the lines intersect?
Lines intersect when they have a point in common. That is, for some x:
Rearranging gives
If two lines do not intersect, then they must be parallel, since the definition of parallel is two straight lines that never meet.
If two lines are the same (possibly in different forms), then their intersection will all real numbers.
Checkpoint
Two lines are plotted on the coordinate plane.
The first line is defined by the point-gradient form equation y−3=−2(x−1). The second line is defined by the standard form equation x−3y+2=0.
Find the point where the two lines intersect.
Select the correct option
There is one "special case" of this kind of intersection. When the gradient m1 of the first line is equal to the negative reciprocal of the gradient m2 of the second line, i.e.
then the point where the lines intersect forms a right angle, and we say they are perpendicular.
Perpendicular Lines
Two lines are perpendicular if they form a right angle with respect to each other. In this case, the rise of one line becomes the run of the other, with a sign change:
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Discussion
Why is it the case that when m1=−m21, two lines form a right angle at their point of intersection?
Why might we care about this?
Two lines with gradients m1 and m2 meet at right angles precisely when one is the negative reciprocal of the other, i.e.
Why is that true? Think about how “rise over run” works:
– On the first line (gradient m1), an increase of Δx=1 produces a rise of Δy=m1. – A line perpendicular to it must turn through 90∘. If you swap the roles of horizontal and vertical moves—and flip one sign—you get a step of
along the second line. Its gradient is
Rewriting gives m1m2=−1, the familiar criterion for perpendicularity in the plane.
Why it matters:
• In triangles, altitudes are perpendicular to the opposite side, so their gradients must be negative‐reciprocals.
• In coordinate geometry, the shortest distance (perpendicular) from a point to a line uses that same fact.
Exercise
Two lines are plotted on the coordinate plane.
The first line is defined by the equation y=x−5. The second is defined by the equation 21x+21y+1=0.
State whether the lines are parallel, the same, or intersect at one point. If they intersect at one point, state whether or not they are perpendicular.
Select the correct option
In finding the points of intersection for these lines, we have actually been going through a process called solving a system of equations! This is an extraordinarily valuable technique and allows us to further ground our abstract study of lines and graphs in useful, real-life considerations.
We've actually already used a lot of these skills. Let's walk through the following problem as an example:
Discussion
Ella is a preschool teacher. She wants to make cookies to bring in to her class. Her goal is to bake a total of 40 cookies, and her boss has given her $75 to spend on supplies. She plans to make some chocolate chip and some oatmeal raisin cookies. It costs about $1.50 in supplies to make each chocolate chip cookie and $2.50 for each oatmeal raisin cookie.
Ella needs to decide how many of each cookie to make in order to both make the right amount and use all the money her boss gave her. Let C be the number of chocolate chip cookies she will make and R the number of oatmeal raisin cookies.
First, set up an equation that describes the total amount of cookies she will make. Remember that she wants to make 40 cookies, with some being chocolate chip and some oatmeal raisin.
Next, set up an equation that relates the total amount of money she wants to spend to the cost of each cookie. Remember that the cost per cookie of the chocolate chip ($1.50) plus the cost per cookie of the oatmeal raisin ($2.50) should add up to a total of $75.
Finally, using the graph below, find how many of each kind of cookie Ella should make. You can slide the values determining each line around to change their equations.
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Part (a)
Let C be the number of chocolate‐chip cookies and R the number of oatmeal‐raisin cookies. Since she wants to bake a total of 40 cookies,
Part (b)
Let C be the number of chocolate‐chip cookies and R the number of oatmeal‐raisin cookies. Since each chocolate‐chip cookie costs $1.50 and each oatmeal‐raisin cookie costs $2.50, their total cost is
Part (c)
On the graph the two lines
meet at their point of intersection. To find that point algebraically, write
Answer: Ella should bake 25 chocolate-chip cookies and 15 oatmeal-raisin cookies.
A line is just a picture of all the points that make one equation "true" (i.e., satisfy the relationship between x and y given by the equation of the line). Solving a system of equations involving two lines means finding a point that makes both equations true at the same time.
In the real world, solving a system is about finding what values make two different conditions true: in the discussion above, for example, the amounts of each cookie where there are 40 cookies overall (the first condition) and the total cost to make them is $75 (the second condition).
We can solve systems of equations graphically or algebraically. To solve them using a graph, sketch the lines given by the equations and identify their point of intersection. To solve them algebraically, use either substitution or elimination.
Systems of equations with 2 unknowns
Suppose we have straight lines with equation 3y+2x−2=0 and 2y−3x+1=0. Where do the lines intersect?
We have the system of equations
There are two ways of solving this.
By substitution
Rearranging
Substituting this into 3y−3x+1=0:
So x=53, which implies y=−32⋅53+32=154. So the intersection is (53,154).
By elimination
We can eliminate y from the equations by subtracting the second from the first:
So x=53⇒y=154 and the intersection is again (53,154).
We can use either of these methods to systems of equations with 2 equations and 2 unknowns.
Exercise
A system of equations is given by
Find the values of x and y that satisfy the system.
Select the correct option
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