To find out if a system of equations has no solutions, we can check if the two equations represent parallel lines when graphed.
For the given system of equations:
4x + 6y = 9
2x + 3y = 6
To convert these equations into slope-intercept form (y = mx + b), we can solve for y in each equation:
Equation 1: 4x + 6y = 9
6y = -4x + 9
y = (-4/6)x + 9/6
y = (-2/3)x + 3/2
Equation 2: 2x + 3y = 6
3y = -2x + 6
y = (-2/3)x + 6/3
y = (-2/3)x + 2
From the slope-intercept form, we can see that both equations have the same slope, which is -2/3. When graphed, two lines with the same slope but different y-intercepts are parallel.
Graphically, the system of equations has no solution because the lines represented by the equations are parallel and will never intersect.
Algebraically, we can also see that there is no solution. If we try to solve the system by setting the equations equal to each other:
(-2/3)x + 3/2 = (-2/3)x + 2
By subtracting (-2/3)x from both sides, we get:
3/2 = 2
This is a contradiction since 3/2 is not equal to 2. Therefore, the system has no solutions.
4x + 6y = 9
2x + 3y = 6
Give a graphical and algebraic explanation as to why the system has no solutions
1 answer
1 answer