Main content

### Course: Algebra 1 > Unit 6

Lesson 5: Number of solutions to systems of equations- Systems of equations number of solutions: fruit prices (1 of 2)
- Systems of equations number of solutions: fruit prices (2 of 2)
- Solutions to systems of equations: consistent vs. inconsistent
- Solutions to systems of equations: dependent vs. independent
- Number of solutions to a system of equations
- Number of solutions to a system of equations graphically
- Number of solutions to a system of equations graphically
- Number of solutions to a system of equations algebraically
- Number of solutions to a system of equations algebraically
- How many solutions does a system of linear equations have if there are at least two?
- Number of solutions to system of equations review

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Number of solutions to a system of equations

Sal is given three lines on the coordinate plane, and identifies one system of two lines that has a single solution, and one system that has no solution. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- Could someone help me understand how to solve a equation then graph it if the equation is in standard form?(29 votes)
- Take it out of standard form and put it in y intercept form then find your cordinate point and graph them that should help you(22 votes)

- Is there any difference between a coordinate plane and a coordinate grid?(17 votes)
- How do you figure out if the problem has infinite, one, or no solutions?(6 votes)
- (I'm going to reference parallel lines multiple times) Assuming you're talking about straight line equations (y=mx+b), if the equations have the same slope, they are either going to have no solutions, or infinite solutions. To picture this, if they have the same slope, but different y-int. they would be parallel lines, which never touch, which is why it has no solutions. Or, if they have the same slope, and same y-int, it would be the same line on top of itself, so every point on one of the lines shows up on the other line, which is why it has infinite solutions. If they have different slopes, then it will only have one solution, opposite from parallel lines, these will intersect once, and only once, which it why it will only have one solution. Sorry if this is too wordy, but hope this helps(18 votes)

- so....correct me if i am wrong...but it seems like the system of two lines that has no solution has the same slope and has different y intercept(8 votes)
- That is correct - that also means that the lines are parallel(8 votes)

- How can I tell if I graphed it right(5 votes)
- there really are no sure signs that you have graphed "right"; just make sure that your incriments are small and equal. If you have graphed right, then the point of intersection that you get should be the coordinate (x,y) points that satisfy your system of equations. hope that helps!(3 votes)

- If I were given a series of equations, say,

2x+2y+4z = 6

3x+6y-5z = 11

Ax+7y-2z = 3 and then asked to give a value of A for which the equations are consistant, how would I go about that? I got this question (not the same equations but the same format) in an assignment (not one that I'm getting any NCEA credits for) and I'm a little lost. I was just wondering if anyone could give me some tips, or point me towards a video that addresses this kind of problem.(6 votes) - what is the difference between solving systems graphically and solving algebraically(4 votes)
- It's just as it sounds, solving systems graphically involves utilizing the coordinate plane to visualize the problem, while if you solve it algebraically you're only using the given algebraic equations and the methods of algebra.(3 votes)

- When it says to use standard form, do you reach standard form by algebraically transferring numbers until y is on the left, and x and whatever other numbers there may be are on the right?(3 votes)
- No that is slope intercept form to isolate the y. Standard form has Ax + By = C, so variables and their coefficients on on left and constants on the right.(2 votes)

- What is the graph that a line represents?0:50-1:00(2 votes)
- A graph of a line is a picture of a linear equation. Every point on the line is a solution to the linear equation.(3 votes)

- How would you figure out the number of solutions for systems of lines in three dimensions?(3 votes)

## Video transcript

We're told to look at the
coordinate grid above. I put it on the side here. Identify one system
of two lines that has a single solution. Then identify one system
of two lines that does not have a solution. So let's do the first part
first. So a single solution. And they say identify one
system, but we can see here there's actually going to
be two systems that have a single solution. And when we talk about a single
solution, we're talking about a single x and y value
that will satisfy both equations in the system. So if we look right here at the
points of intersection, this point right there, that
satisfies this equation y is equal to 0.1x plus 1. And it also satisfies, well,
this blue line, but the graph that that line represents,
y is equal to 4x plus 10. So this dot right here, that
point represents a solution to both of these. Or I guess another way to think
about it, it represents an x and y value that satisfy
both of these constraints. So one system that has one
solution is the system that has y is equal to 0.1x plus 1,
and then this blue line right here, which is y is equal
to 4x plus 10. Now, they only want us to
identify one system of two lines that has a single
solution. We've already done that. But just so you see it, there's
actually another system here. So this is one system right
here, or another system would be the green line and
this red line. This point of intersection right
here, once again, that represents an x and y value
that satisfies both the equation y is equal to 0.1x plus
1, and this point right here satisfies the equation
y is equal to 4x minus 6. So if you look at this system,
there's one solution, because there's one point of
intersection of these two equations or these two lines,
and this system also has one solution because it has one
point of intersection. Now, the second part of the
problem, they say identify one system of two lines that does
not have a single solution or does not have a solution,
so no solution. So in order for there to be no
solution, that means that the two constraints don't overlap,
that there's no point that is common to both equations or
there's no pair of x, y values that's common to
both equations. And that's the case of the two
parallel lines here, this blue line and this green line. Because they never intersect,
there's no coordinate on the coordinate plane that satisfies
both equations. So there's no x and y
that satisfy both. So the second part of the
question, a system that has no solution is y is equal to 4x
plus 10, and then the other one is y is equal
to 4x minus 6. And notice, they have the exact
same slope, and they're two different lines, they have
different intercepts, so they never, ever intersect,
and that's why they have no solutions.