Solve the system using elimination.
3x - 5y = 18
4x + 6y = -14
(Simplify your answer. Type an ordered pair.)
To solve the system using elimination, we want to eliminate one variable by multiplying one or both of the equations by a constant so that when we add or subtract the two equations, one of the variables will cancel out.
To begin, let us multiply the first equation by 4 and the second equation by 3:
4(3x - 5y) = 4(18)
3(4x + 6y) = 3(-14)
Expanding these expressions, we get:
12x - 20y = 72
12x + 18y = -42
Now, we can subtract the second equation from the first to eliminate the variable 'x':
(12x - 20y) - (12x + 18y) = 72 - (-42)
12x - 12x - 20y - 18y = 72 + 42
-38y = 114
To isolate 'y', we divide both sides of the equation by -38:
y = -3
Now, substituting the value of 'y' back into either of the original equations, let's use the first equation:
3x - 5(-3) = 18
3x + 15 = 18
Subtracting 15 from both sides:
3x = 3
Simplifying further:
x = 1
Therefore, the solution to the system of equations is the ordered pair (x, y) = (1, -3).
Tell whether the system has one solution, infinitely many solutions, or no solution.
2x + 3y = 14
2x + 3y = 42
Choose the correct answer below.
A.
The system has infinitely many solutions.
B.
The system has no solution.
C.
The system has one solution.
To determine whether the system has one solution, infinitely many solutions, or no solution, we can examine the coefficients of both equations. If the coefficients of both variables (x and y) are the same in both equations, and the constants on the right side of the equations are different, then the system has no solution.
Looking at the system of equations:
2x + 3y = 14
2x + 3y = 42
We can see that the coefficients of both variables (2x and 3y) are the same in both equations. However, the constants on the right side of the equations (14 and 42) are different.
Thus, we can conclude that the system has no solution, and the correct answer is B. The system has no solution.
Tell whether the system has one solution, infinitely many solutions, or no solution.
-x + 2y = -3
4x - 8y = 12
Choose the correct answer below.
A.
The system has one solution.
B.
The system has infinitely many solutions.
C.
The system has no solution.
To determine whether the system has one solution, infinitely many solutions, or no solution, we can examine the coefficients of both equations.
Looking at the system of equations:
-x + 2y = -3
4x - 8y = 12
We can see that the coefficients of both variables (x and y) are different in both equations. Therefore, the system may have one solution.
To confirm if the system has a unique solution, let's solve the system using the elimination method:
Multiply the first equation by 4 to eliminate the x variable:
4(-x + 2y) = 4(-3)
-4x + 8y = -12
Now, let's add this equation to the second equation:
(-4x + 8y) + (4x - 8y) = -12 + 12
0 = 0
The result is 0 = 0, which means that both equations are equivalent. This indicates that the system has infinitely many solutions.
Therefore, the correct answer is B. The system has infinitely many solutions.
Solve the system using any method.
x + y = 4.4
2x + y = 3.8
What is the solution of the system of equations?
To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of elimination:
Multiply the first equation by 2 to match the coefficients of y in both equations:
2(x + y) = 2(4.4)
2x + 2y = 8.8
Now, subtract the second equation from this new equation to cancel out the y variable:
(2x + 2y) - (2x + y) = 8.8 - 3.8
x + y = 5
Now we have a new simplified equation:
x + y = 5
Next, we can solve for one variable in terms of the other. Let's solve for y:
y = 5 - x
Now, we can substitute this expression for y into either of the original equations. Let's use the first equation:
x + (5 - x) = 4.4
Simplifying:
x + 5 - x = 4.4
5 = 4.4
This is not a true statement, so the system of equations is inconsistent and has no solution.
Therefore, the correct answer is: The system of equations has no solution.