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### Course: Algebra (all content)>Unit 5

Lesson 2: Introduction to systems of equations

# Systems of equations with graphing: exact & approximate solutions

Sal solves a system of two linear equations in standard form, and then approximates the solution of a system whose solution isn't clearly visible.

## Want to join the conversation?

• At , Khan says that when y is zero, x is negative one. I don't get how he got that. Can anyone explain this in another way?
• Sal is now plotting points that lie on the line defined by the second equation in the system of equations. the relation he is graphing is 6x - 6y = -6. One way to plot a line is to plot any two points that are on the line, and for an equation in standard from like this one, two easy points to find at the x and y intercepts--the values where x=0 and y=0. To find the x intercept, plug y=0 into 6x-6y=-6 and you get 6x-6(0)=-6 which simplifies to 6x=-6 or x=-1. Similarly, to find the y-intercept, let x=0. you get 6(0)-6y=6 which simplifies to -6y=-6 or y=1. So the points (-1,0) and (0,1) are on our line.

You can see a video explaining the process of finding intercepts

• I didn't understand the concept
• Pretty much, the concept is to replace the x and y values with simple values and graph the result. In the case of a fraction like 1/3X, the most simple value would probably be 3 as it would remove the fraction. The X and Y values are not set in stone for substitution.
• How much of a lesson should I get done each day on khan academy?
• Normally, I try to complete one of the lesson boxes, as in the videos or exercises in one of the rectangles in the unit. It could just be a quiz that day or a unit test. Either that or I give myself an hour to do as much as possible.
• I don't understand on how he found not the y-intercept, but the other part on graphing.
• At , Sal just estimated the approximate numbers but are there any ways to get the exact numbers using more mathematical ways?
• Yes, there are other methods. You will soon learn them in Algebra 1:
-- Substitution method
-- Elimination or addition method

There is another method that uses matrices that is taught in PreCalculus.
• what about if the equation is in slope intercept form and the slope is a whole integer?
• If both the slope and 𝑦-intercept are integers,
we can plug 𝑥 = 0 ⇒ 𝑦 = 𝑏 and 𝑥 = 1 ⇒ 𝑦 = 𝑚 + 𝑏

Example: 𝑦 = 7𝑥 − 5
𝑥 = 0 ⇒ 𝑦 = −5
𝑥 = 1 ⇒ 𝑦 = 7 − 5 = 2
So, (0, −5) and (1, 2) are two points on the line.
• I am so confused. Next time, could you explain slower?
• 1) You can slow down the video yourself. Click on the gear symbol in the lower right of the video window and adjust the video speed.
2) Use the pause button as soon as you start to get confused. Try reviewing the transcript to see if that helps eliminate your confusion.
3) You can watch the video as many times as you need to.
• So,like, I don't get this. For example, when Sal says that "When x is equal to zero, y would be equal to negative three" what does that mean?
(1 vote)
• As Sal states - He is picking different values of X and then calculating Y using one of the equations. In the one you referenced, Sal is using the first equation: -x-3y=9. If you use x=0, the equation becomes: -0-3y=9, then solve for Y.
-3y=9
y=-3
This creates one point for graphing the first line. The point is (0, -3). Sal repeats this process using other values of X to find 2 points for each line.

Hope this helps.
• So how do you plot the dot when, for example, x=2/3?
• Fractions sit in between the integers on a number line.
For x=2/3, it is located between 0 and 1. Split the space between 0 and 1 into 3 equal sections. 2/3 is 2 of those sections from 0.

Search for the lesson on "fractions on a number line" for more details.
• So, how would you plot something like
7x−y=7

x+2y=6
​ I cannot figure out how to plot it.
• You have a couple of options:
1) You can convert each equation to slope-intercept form, then graph using the y-intercept and the slope.

2) You can calculate 2 points for each line. Once you have 2 points for the line, you can draw the line. To find a point, pick a value for X or Y and put it into the equation. Then, calculate the other variable. For example: if y=0
7x-0=7
7x=7
x=1
You now have the point (1,0) that can be graphed.

Hope this helps.