solve each system of equations on the left. Match the system to the correct number of solutions
y=3x-2. 2x+3y=16. (one solution, infinite solutions, or no solutions?)
2x-2y+-4. y=x+2. (one solution, infinite solutions, or no solutions?)
y=-2x-7. y=-2x+3. (one solution, infinite soulutions, or no solutions?)
For the first system of equations:
To solve this system of equations, we can use the method of substitution. We already have the equation y = 3x - 2. We can substitute this expression for y in the second equation and solve for x:
2x + 3(3x - 2) = 16
2x + 9x - 6 = 16
11x = 22
x = 2
Now, substitute the value of x back into the first equation to solve for y:
y = 3(2) - 2
y = 6 - 2
y = 4
Therefore, the first system of equations has one solution.
For the second system of equations:
To solve this system of equations, we can set the two equations equal to each other:
2x - 2y - 4 = x + 2
Now, solve for x:
2x - x = 2y + 2 + 4
x = 2y + 6
Substitute this expression for x into the second equation:
y = 2y + 6 + 2
y = 2y + 8
Rearrange this equation:
0 = y + 8 - 2y
0 = -y + 8
y = 8
Now, substitute the value of y back into the expression for x:
x = 2(8) + 6
x = 22
Therefore, the second system of equations has one solution.
For the third system of equations:
The two equations are y = -2x - 7 and y = -2x + 3. These equations have the same slope (-2), but different y-intercepts (-7 and 3). When two lines have the same slope and different y-intercepts, they are parallel and will never intersect. Therefore, the third system of equations has no solutions.