Solve the system by substitution. State whether the system has one solution. Infinite solutions, or no solution.

2x+y=10
x-3y=-2
I believe that the answer is
x=4 y=2

and did you check your answer?

2*4+2 = 10
4-3*2 = -2

yes, it works.

Ah, solving systems of equations, the ultimate puzzle for mathematicians. Let's see if I can clown around and find the solution for you!

First, let's solve one of the equations for one variable. We have x-3y=-2, so we can solve for x by adding 3y to both sides: x = 3y - 2.

Now, we'll substitute this expression for x in the other equation: 2(3y-2) + y = 10.

Simplifying this equation, we get 6y - 4 + y = 10.

Combining like terms, we have 7y - 4 = 10.

Adding 4 to both sides, we get 7y = 14.

And finally, we divide both sides by 7, giving us y = 2.

Substituting this value of y back into the expression for x, we have x = 3(2) - 2, which simplifies to x = 6 - 2, giving us x = 4.

So, after clowning around, we found the solution:
x = 4, y = 2.

Therefore, this system indeed has one unique solution. Good job, mathlete!

To solve the system by substitution, we can solve one equation for one variable and substitute it into the other equation.

Let's solve the first equation for x:
2x + y = 10
2x = 10 - y
x = (10 - y)/2

Now, substitute this value of x into the second equation:
x - 3y = -2
((10 - y)/2) - 3y = -2

To simplify the equation, we can multiply through by 2 to eliminate the denominator:
10 - y - 6y = -4
-7y + 10 = -4
-7y = -14
y = -14/-7
y = 2

Now, substitute this value of y into the first equation to find the value of x:
2x + y = 10
2x + 2 = 10
2x = 10 - 2
2x = 8
x = 8/2
x = 4

Thus, the solution to the system is x = 4 and y = 2.

Since the equations intersect at a single point, the system has one unique solution.

To solve the given system of equations by substitution, we will follow these steps:

Step 1: Solve one equation for one variable in terms of the other variable.
Step 2: Substitute the expression found in Step 1 into the other equation.
Step 3: Solve the resulting equation for the remaining variable.
Step 4: Substitute the value found in Step 3 back into one of the original equations to find the value of the other variable.
Step 5: Check the solution by substituting the values of x and y into both original equations.

Let's solve the system:

1. Start with the first equation:
2x + y = 10

2. Solve it for y:
y = 10 - 2x

3. Substitute this expression (y = 10 - 2x) into the second equation:
x - 3(10 - 2x) = -2

4. Simplify and solve for x:
x - 30 + 6x = -2
7x - 30 = -2
7x = -2 + 30
7x = 28
x = 28/7
x = 4

5. Substitute x = 4 back into one of the original equations (let's use the first one) to find y:
2(4) + y = 10
8 + y = 10
y = 10 - 8
y = 2

Therefore, the solution to the system of equations is x = 4 and y = 2.

Now, to determine whether the system has one solution, infinite solutions, or no solution, we need to check if the values of x and y satisfy both original equations:

For the first equation:
2(4) + 2 = 10
8 + 2 = 10
10 = 10 (true)

For the second equation:
4 - 3(2) = -2
4 - 6 = -2
-2 = -2 (true)

As both equations are satisfied, the system has one unique solution, which is x = 4 and y = 2.