4x + 6y = 9
2x + 3y = 6
Give a graphical and algebraic explanation as to why the system has no solutions
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.
To determine if a system of equations has no solution, we can examine the graphical representation and perform algebraic manipulation.
Graphical Explanation:
To graph the two equations, we need to convert them into slope-intercept form, which is in the form of y = mx + b.
Equation 1: 4x + 6y = 9
Rearranging, we have 6y = -4x + 9.
Dividing both sides by 6, we get y = (-4/6)x + 9/6, which simplifies to y = (-2/3)x + 3/2.
Equation 2: 2x + 3y = 6
Rearranging, we have 3y = -2x + 6.
Dividing both sides by 3, we get y = (-2/3)x + 2.
Both equations have the same coefficient (-2/3) for x. This means they have the same slope. However, the y-intercepts (3/2 and 2) are different.
When we plot the two lines on a graph, we will find that they are parallel since they have the same slope but different y-intercepts. Parallel lines do not intersect, which implies that there is no common solution for the system of equations.
Algebraic Explanation:
We can also show algebraically why the system has no solution by performing operations on the equations.
We can solve the first equation for x in terms of y:
4x + 6y = 9
4x = 9 - 6y
x = (9 - 6y) / 4
Substituting this into the second equation:
2x + 3y = 6
2[(9 - 6y) / 4] + 3y = 6
(9 - 6y) / 2 + 3y = 6
(9 - 6y) + 6y = 12
9 = 12
The equation simplifies to 9 = 12, which is false. This contradiction indicates that the system of equations has no solution.
In conclusion, both the graphical and algebraic explanations show that the system of equations has no solutions because the lines are parallel and do not intersect.