solve the following system of equations
2 (square root x) + 5y=16
6 (square root x) - 3y=12
To solve the system of equations, we need to eliminate the square root term. Here's how you can do it step-by-step:
1. Let's first eliminate the square root term. To do that, we will square both sides of both equations.
(2 √x + 5y)^2 = 16^2
(6 √x - 3y)^2 = 12^2
Simplifying this will give us:
4x + 20√x y + 25y^2 = 256 ...(Equation 1)
36x - 36√x y + 9y^2 = 144 ...(Equation 2)
2. Next, we want to eliminate the 'y' term, so we can solve for 'x'. To accomplish this, we will multiply Equation 1 by 9 and Equation 2 by -25 to make the coefficients of 'y^2' equal. This will allow us to subtract equation 2 from equation 1, effectively eliminating the 'y^2' term.
36x + 180√x y + 225y^2 = 2304 ...(Equation 1 multiplied by 9)
-36x + 36√x y - 9y^2 = -360 ...(Equation 2 multiplied by -25)
Adding these two equations together will give us:
-9√x y + 216y^2 = 1944
3. Now, we have a quadratic equation in terms of 'y', which we can solve to find the value of 'y'. Rearranging the equation, we get:
216y^2 - 9√x y = 1944
Factoring out 'y' common, we have:
y(216y - 9√x) = 1944
Dividing both sides by 9, we get:
24y(24y - √x) = 216
Simplifying further:
576y^2 - 24y√x - 216 = 0
4. This equation is quadratic in terms of 'y'. We can use the quadratic formula to solve for 'y':
y = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 576, b = -24√x, and c = -216.
Substituting these values into the quadratic formula, we have:
y = (-(-24√x) ± √((-24√x)^2 - 4 * 576 * -216)) / (2 * 576)
y = (24√x ± √(576x + 186624)) / 1152
y = (24√x ± √(576x + 186624)) / 1152
5. Now that we have the value of 'y', we can substitute it back into one of the original equations to solve for 'x'. Let's substitute it into the first equation:
2 √x + 5(24√x ± √(576x + 186624)) / 1152 = 16
Simplify this equation and solve for 'x'. Once you have the value of 'x', substitute it back to find the value of 'y'.
Note: This process can be a bit complex, so you may need a calculator to find the values of 'x' and 'y'.
To solve the given system of equations, which contains square roots of x, we will use a substitution method. Here's the step-by-step process:
Step 1: Let's start by solving one of the equations for one variable in terms of the other variable. We'll solve the first equation for square root x.
2 (square root x) + 5y = 16
Subtract 5y from both sides:
2 (square root x) = 16 - 5y
Divide both sides by 2:
(square root x) = (16 - 5y) / 2
Step 2: Now, we'll substitute the expression we found for square root x in the second equation and solve for y:
6 (square root x) - 3y = 12
Substitute (16 - 5y) / 2 for square root x:
6 [(16 - 5y) / 2] - 3y = 12
Simplify by multiplying both sides by 2 to get rid of the fraction:
6 (16 - 5y) - 6y = 24
Distribute 6:
96 - 30y - 6y = 24
Combine like terms:
-36y = -72
Divide both sides by -36:
y = -72 / -36
y = 2
Step 3: Now that we know the value of y, we can substitute it back into one of the original equations to find the value of square root x.
Using the first equation:
2 (square root x) + 5y = 16
Substitute y = 2:
2 (square root x) + 5(2) = 16
2 (square root x) + 10 = 16
Subtract 10 from both sides:
2 (square root x) = 6
Divide both sides by 2:
(square root x) = 6 / 2
(square root x) = 3
Now we have the value of square root x.
Step 4: To find the value of x, we need to square both sides of the equation:
(square root x) = 3
(x) = 3^2
(x) = 9
So, the solution to the system of equations is x = 9 and y = 2.
Let's solve the system of equations:
First, let's solve for the square root of x:
From the first equation, we have:
2(sqrt(x)) + 5y = 16
Subtracting 5y from both sides:
2(sqrt(x)) = 16 - 5y
Dividing both sides by 2:
sqrt(x) = (16 - 5y)/2
Now, let's square both sides to eliminate the square root:
x = ((16 - 5y)/2)^2
x = (8 - 5y/2)^2
x = (64 - 40y + 25y^2/4)
Now, let's substitute this value of x into the second equation:
6(sqrt(x)) - 3y = 12
Substituting the expression for x:
6(sqrt((64 - 40y + 25y^2)/4)) - 3y = 12
Simplifying:
6(sqrt(16 - 10y + 25y^2)/2) - 3y = 12
Multiplying both sides by 2:
6(sqrt(16 - 10y + 25y^2)) - 6y = 24
Squaring both sides to eliminate the square root:
36(16 - 10y + 25y^2) - 72y(16 - 10y + 25y^2) + 36y^2 = 576
Expanding and simplifying:
576 - 360y + 900y^2 - 1152y + 720y^2 - 180y^3 + 36y^2 = 576
Combining like terms:
-180y^3 + 1656y^2 - 1512y = 0
Dividing through by -12y:
15y^2 - 138y + 126 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
The factored form of the equation is:
(3y - 6)(5y - 21) = 0
Setting each factor equal to zero and solving for y:
3y - 6 = 0 --> 3y = 6 --> y = 2
5y - 21 = 0 --> 5y = 21 --> y = 21/5
Therefore, there are two potential solutions for y: y = 2 and y = 21/5.
Now, let's substitute the values of y back into the original equations to find the corresponding values of x:
For y = 2:
2(sqrt(x)) + 5(2) = 16
2(sqrt(x)) + 10 = 16
2(sqrt(x)) = 6
sqrt(x) = 3
x = 9
For y = 21/5:
2(sqrt(x)) + 5(21/5) = 16
2(sqrt(x)) + 21 = 16
2(sqrt(x)) = -5
This equation has no real solutions, since the square root of a number cannot be negative.
Therefore, the solution to the system of equations is x = 9 and y = 2.