Tell whether the system has one solution infinitely many solutions or no solution.
1.5x+2y=11
3x+6y=22
a. one solution
b. infinitely many solutions
c. no solution
Multiply Eq1 by -2 and add:
Eq1: -3x - 4y = -22.
Eq2: +3x + 6y = 22.
Sum: 2y = 0.
Y = 0.
In Eq2, replace y with 0 and solve for x:
3x + 6*0 = 22.
X = 7 1/3.
One solution.
To determine whether the system has one solution, infinitely many solutions, or no solution, we can solve the system using the method of elimination.
Given the system of equations:
1. 5x + 2y = 11
2. 3x + 6y = 22
To eliminate y, we can multiply equation 1 by 3 and equation 2 by -5:
3 * (5x + 2y) = 3 * 11
-5 * (3x + 6y) = -5 * 22
Which simplifies to:
15x + 6y = 33
-15x - 30y = -110
Adding both equations together, we get:
(15x + 6y) + (-15x - 30y) = 33 + (-110)
6y - 30y = -77
-24y = -77
Simplifying further, we find:
y = -77 / -24
y = 3.208
Substituting the value of y back into one of the original equations (equation 1), we have:
5x + 2(3.208) = 11
5x + 6.416 = 11
5x = 4.584
x = 4.584 / 5
x = 0.9168
Therefore, the system has one solution (x = 0.9168, y = 3.208).
Hence, the answer is a. one solution.
To determine whether the system has one solution, infinitely many solutions, or no solution, you can use the method of elimination or substitution.
1. Method of Elimination:
First, we'll multiply both sides of the first equation by 3 to make the coefficients of x the same in both equations:
(3)(1.5x + 2y) = (3)(11)
4.5x + 6y = 33
Now, we can subtract the second equation:
(4.5x + 6y) - (3x + 6y) = 33 - 22
4.5x + 6y - 3x - 6y = 11
(4.5x - 3x) + (6y - 6y) = 11
1.5x + 0 = 11
1.5x = 11
Since the variable x remains in the equation, it means there is only one value for x that satisfies this equation. Therefore, this system has one solution (a).
2. Method of Substitution:
We'll solve one equation for one variable and substitute it into the other equation.
Let's solve the first equation for x:
1.5x + 2y = 11
1.5x = 11 - 2y
x = (11 - 2y)/1.5
Now, we'll substitute this value into the second equation:
3((11 - 2y)/1.5) + 6y = 22
Simplifying this equation, we get:
(33 - 6y)/1.5 + 6y = 22
33 - 6y + 9y = 33
3y = 0
The variable y simplifies to 0, which means the value of y is 0. Substituting this back into x = (11 - 2y)/1.5, we get:
x = (11 - 2(0))/1.5
x = 11/1.5
x = 22/3
Since the variables x and y have specific values, this system has one solution (a).
In conclusion, both the method of elimination and substitution led to the conclusion that this system has one solution (a).