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### Course: 8th grade (Illustrative Mathematics) > Unit 4

Lesson 10: Lesson 12: Systems of equations- Systems of equations: trolls, tolls (1 of 2)
- Systems of equations: trolls, tolls (2 of 2)
- Systems of equations with graphing
- Systems of equations with graphing
- Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
- Systems of equations with graphing
- Number of solutions to a system of equations graphically

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# Systems of equations with graphing

Walk through examples of solving systems of equations by finding the point of intersection.

We can find the solution to a system of equations by graphing the equations. Let's do this with the following systems of equations:

First, let's graph the first equation ${y={\displaystyle \frac{1}{2}}x+3}$ . Notice that the equation is already in $y$ -intercept form so we can graph it by starting at the $y$ -intercept of $3$ , and then going up $1$ and to the right $2$ from there.

Next, let's graph the second equation ${y=x+1}$ as well.

There is exactly one point where the graphs intersect. This is the solution to the system of equations.

This makes sense because every point on the gold line is a solution to the equation ${y={\displaystyle \frac{1}{2}}x+3}$ , and every point on the green line is a solution to ${y=x+1}$ . So, the only point that's

**a solution to both equations**is the point of intersection## Checking the solution

So, from graphing the two equations, we found that the ordered pair $(4,5)$ is the solution to the system. Let's verify this by plugging $x=4$ and $y=5$ into each equation.

The first equation:

The second equation:

Nice! $(4,5)$ is indeed a solution.

## Let's practice!

### Problem 1

### Problem 2

### Problem 3

## Challenge problems

## Want to join the conversation?

- Is there some potion or something I can drink to become better and have MEMORY RETENTION when it comes to math? I am in college and this is all new to me. I moved around alot when younger, so I never really got the concept of more advanced mathematics. I just stare at problems for long times and have no idea where to start half the time, but once I get rolling I don't stop. Any advice?(50 votes)
- How could you convert a normal system of equations into slope intercept form?(16 votes)
- You've waited 5 years for the answer, so here it is. You can solve one equation for either variable, then plug that into the other equation and solve that one completely, plug that back into the first equation, and now you know all the variables and you can do whatever you want with them, like put them into slope intercept form.(27 votes)

- i litterally have no clue what im doing rn!(18 votes)
- I love Khan Academy because it's the only thing thatś actually helping me learn this stuff right now but The US education system is truly a failure.(19 votes)

- Please help me! I have no idea how to even find out which coordinates I am supposed to put these lines on. They don't explain this concept well.(17 votes)
- When given a Slope-Intercept form equation what you can do to graph is graph the y-intercept "b" which is the number without a variable on the graph first, this means you have to put that point on the vertical line where the "b" is shown. Then you just have to move one on the x-axis (the horizontal) and the amount next to the x up or down (the number next to the x is the "m" better known as the slope). I hope this helps.(6 votes)

- Is there anyone who can explain this to me?

I'm in 8th grade. My teacher uses a service called "my.hrw" and every time you get it wrong, it never explains why. I know adults say "ask the teacher" but he always explains like I know what he's saying. There are lots of things about this and all the adults I ask about math in general get angrier every time I ask a question like "ok, but why are we doing this" or "why do we use**THIS**number instead of**THAT**number." I've asked the teacher to explain it to me multiple times, I've watched almost every YouTube and Khan Academy video at this point, and looked on every website imaginable. It seems like everyone else gets this concept except I. I've got to learn this to get to the next thing that I'm supposed to be learning right now but I can't grasp that concept because it requires you know how to do this! Can*ANYONE*explain this in simple, easy to understand terms, and break it down step by step, process by process?(17 votes)- darn thats a lot of confusion. concept like this are hard but the best piece of advice I can give is focus on how each piece of the problem connect, break it down if you have to, and try and piece it together in a way you may be able you understand so things like these won't cause you so much trouble, like if you look at y = 1/2x + 3, I notice that in the graph it uses the 1/2 to make the angle that the line is at, right? then I observe that the line goes up one on the y-axis, and right two on the x-axis, which makes 1/2 which are both positive numbers.(7 votes)

- I hope everyone on Khan academy has a good day and a wonderful life! :)(17 votes)
- it was really hard to understand(14 votes)
- *If I am being frank, this is very hard, I took around 2-3 hours of work to understand. Just keep on going.*(4 votes)

- I dont quite understand the graphing part, what exactly do we have do in order to get the numbers? Sorry im a slow learner.(10 votes)
- You can graph any line by calculating two points on the line. If you want to ensure that you have good points, then you would also calculate a 3rd point. If you can line up 3 points in a straight line, you likely don't have errors.

Some ways to find points:

1) Pick a value for X and use the equation to calculate the corresponding Y value. You then have one point (x, y). See examples at: https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8th-solutions-to-two-var-linear-equations/v/graphing-solutions-to-2-variable-linear-equations-1

2) Find the X and Y intercepts. You can learn how to do this at: https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8th-x-and-y-intercepts/v/introduction-to-intercepts

3) If the equation is in slope intercept form: y=mx+b, you are given a point, the y-intercept at (0, b). Graph that point, then use the slope (m) to find other points. You can learn about this at: https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8th-slope-intercept-form/v/graphing-a-line-in-slope-intercept-form

Note: The lesson on systems of linear equations assumes that you know these skills. So, you may want to brush up on them.

Hope this helps.(10 votes)

- How does this make any sense?(11 votes)
- can a system of equations be be in multiple ways(9 votes)
- Yes, the two main ways include elimination and substitution. There are other methods as well including using graphs and matrix.

This might help you: https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-systems-topic/cc-8th-systems-solutions/v/inconsistent-systems-of-equations(5 votes)