solve following using "elimination method"
3x^2 + 4y = 17
2x^2 + 5y = 12
Multiply the first equation by 2 and the second equation by 3, and subtract one from the other. That will eliminate the x^2 variable.
6x^2 + 8y = 34
6x^2 + 15y = 36
7 y = 2
y = 2/7
2x^2 = 12 - 5y = 12 - 10/7
= 74/7
x = sqrt(37/7) = 2.299
To solve the system of equations using the elimination method, we need to eliminate one variable by multiplying the equations by suitable constants and then adding them together. Let's start:
1. Multiply the first equation by 2 and the second equation by 3 to make the coefficients of x^2 terms the same:
Equation 1: 2(3x^2 + 4y) = 2(17) => 6x^2 + 8y = 34
Equation 2: 3(2x^2 + 5y) = 3(12) => 6x^2 + 15y = 36
2. Subtract the second equation from the first equation to eliminate x^2:
(6x^2 + 8y) - (6x^2 + 15y) = 34 - 36
6x^2 - 6x^2 + 8y - 15y = -2
-7y = -2
3. Divide both sides of the equation by -7 to solve for y:
-7y / -7 = -2 / -7
y = 2/7
4. Let's substitute the value of y into one of the original equations. Let's use Equation 1:
3x^2 + 4(2/7) = 17
3x^2 + 8/7 = 17
5. Subtract 8/7 from both sides of the equation:
3x^2 = 17 - 8/7
3x^2 = (17 * 7 - 8) / 7
3x^2 = 111/7
x^2 = 111/7 / 3
x^2 = 37/7
6. Take the square root of both sides to solve for x:
x = ±√(37/7)
Therefore, the solutions to the system of equations are:
x = ±√(37/7)
y = 2/7
To solve the system of equations using the elimination method, we need to eliminate one of the variables by multiplying both equations by appropriate factors.
Let's start by trying to eliminate the variable y. To do this, we need to find a common multiple of the coefficients of y in both equations. In this case, the coefficients are 4 and 5. The least common multiple of 4 and 5 is 20.
To eliminate y, we'll multiply the first equation by 5 and the second equation by -4, so that the coefficients of y will be opposites:
5(3x^2 + 4y) = 5(17)
-4(2x^2 + 5y) = -4(12)
This simplifies the system of equations to:
15x^2 + 20y = 85
-8x^2 - 20y = -48
Now, we can eliminate y by adding the equations together:
(15x^2 + 20y) + (-8x^2 - 20y) = 85 + (-48)
7x^2 = 37
Divide both sides of the equation by 7 to isolate x^2:
x^2 = 37/7
Now, we can determine the value of x by taking the square root of both sides of the equation:
x = ± sqrt(37/7)
Next, substitute the value of x back into one of the original equations to solve for y. Let's use the first equation:
3x^2 + 4y = 17
For x = sqrt(37/7):
3(sqrt(37/7))^2 + 4y = 17
3(37/7) + 4y = 17
(111/7) + 4y = 17
4y = 17 - (111/7)
4y = (119 - 111)/7
4y = 8/7
y = (2/7)
For x = -sqrt(37/7):
3(-sqrt(37/7))^2 + 4y = 17
3(37/7) + 4y = 17
(111/7) + 4y = 17
4y = 17 - (111/7)
4y = (119 - 111)/7
4y = 8/7
y = (2/7)
Therefore, there are two solutions to the system of equations: (sqrt(37/7), 2/7) and (-sqrt(37/7), 2/7).