Tell whether the system has one solution, infinitely many solutions, or no solution.

1.) y= 1/2x +3
2y -x =6

2.) x= -7y + 34
x+ 7y =32

2 y = x + 6

2 y = x + 6
identical, same line, infinite solutions

x + 7 y = 34
x + 7 y = 32
parallel lines, never cross, no solution

Thanks again! :)

You are welcome again :)

To determine whether a system of equations has one solution, infinitely many solutions, or no solution, we need to solve the system and analyze the result.

1.) y = 1/2x + 3
2y - x = 6

First, let's solve the system. We can use the method of substitution or elimination to find the values of x and y that satisfy both equations.

Using substitution:
Step 1: Substitute the expression for y from the first equation (y = 1/2x + 3) into the second equation:
2(1/2x + 3) - x = 6
x + 6 - x = 6
6 = 6

When solving for x, we find that the equation simplifies to 6 = 6. This indicates that no matter what value of x we substitute, the equation will always be true. This suggests that the system has infinitely many solutions since any x-value will satisfy it.

Using elimination:
Step 1: Multiply the first equation by 2 to get rid of the fraction:
2y = x + 6

Step 2: Rewrite the second equation in standard form:
2y - x = 6

Step 3: Combine the expressions for 2y, equating the coefficients of x:
x + 6 - x = 6
0 = 0

Again, when solving for x, we find that the equation simplifies to 0 = 0. This also suggests that the system has infinitely many solutions.

Hence, the system of equations 1.) y = 1/2x + 3 and 2y - x = 6 has infinitely many solutions.

Now, let's move on to the second system of equations:

2.) x = -7y + 34
x + 7y = 32

Again, we'll solve this system using either substitution or elimination:

Using substitution:
Step 1: Substitute the expression for x from the first equation (x = -7y + 34) into the second equation:
(-7y + 34) + 7y = 32
34 = 32

In this case, we find that 34 = 32, which is not true. This means that the system of equations is inconsistent and has no solution.

Using elimination:
Step 1: To cancel out the x terms, multiply the second equation by -1:
-x - 7y = -32

Step 2: Add the two equations together:
(-7y + 34) + (-x - 7y) = 32 - 32
-14y + 34 - 32 = 0
-14y + 2 = 0
-14y = -2
y = 1/7

Step 3: Substitute the y-value into one of the original equations, let's use equation 1:
x = -7(1/7) + 34
x = -1 + 34
x = 33

We find the unique solution (x, y) = (33, 1/7).

Hence, the system of equations 2.) x = -7y + 34 and x + 7y = 32 has only one solution.