Answer: The given system of equations is a linear system in two variables and can be solved either by substitution or elimination method.
Substitution method:
Solve one of the equations for one of the variables and substitute it into the other equation.
Starting with the first equation:
y = 3x - 7
Substitute this into the second equation:
6x - 2(3x - 7) = 12
Expanding the right side:
6x - 6x + 14 = 12
Simplifying:
14 = 12
This is not a valid solution, so the system has no solution.
Elimination method:
Multiply one of the equations by a constant to make the coefficients of one of the variables equal, then add the two equations to eliminate that variable.
Starting with the first equation:
y = 3x - 7
Multiply the second equation by -2:
-12x + 4y = -24
Adding the two equations:
4y - 12x = -24 - (-7)
Expanding the right side:
4y - 12x = -17
Solving for y:
y = (-12x - 17) / 4
Substituting this expression for y back into either equation to solve for x:
y = 3x - 7
(-12x - 17) / 4 = 3x - 7
Expanding both sides:
-12x - 17 = 4(3x - 7)
Expanding the right side:
-12x - 17 = 12x - 28
Adding 12x to both sides:
-17 = 24x - 28
Adding 28 to both sides:
11 = 24x
Dividing both sides by 24:
x = 11/24
Substituting this value of x back into either equation to find y:
y = 3x - 7
y = 3(11/24) - 7
Expanding:
y = 11/8 - 7
y = 11/8 - 56/8
y = -45/8
So, the solution to the system of equations is (x, y) = (11/24, -45/8).
This solution can be plotted on the coordinate plane to graph the two lines. The two lines intersect at the point (11/24, -45/8), which is the solution to the system of equations.
Explanation: