Solve the system of equations by an algebraic method. Leave answer in exact form.

4x² - 7y² = 21
x² + 2y² = 99

4x² - 7y² = 21 (eq1)

x² + 2y² = 99 (eq2)

--> eq1-4*eq2
-->
4x² - 7y² = 21 (eq1)
-4x² - 8y² =- 396 -4*(eq2)
___________________

-15y²=-375 (eq3)

--> y²=25 (eq4)

Back sub to get
x² + 2(25) = 99
--> x² =99-50
--> x² =49 (eq5)

x =(+/-)sqrt(49) (eq6)

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# x =(+/-)7 # (eq7)
###############

y =(+/-)sqrt(25) (eq8)

###############
# y =(+/-)5 # (eq9)
###############

You could also use Matrix Methods

To solve the system of equations algebraically, we can use the method of substitution. We'll solve one equation for one variable and substitute it into the other equation. Let's solve the second equation for x².

x² + 2y² = 99

Subtract 2y² from both sides:

x² = 99 - 2y²

Now, substitute this expression for x² in the first equation:

4(99 - 2y²) - 7y² = 21

Expand and simplify:

396 - 8y² - 7y² = 21

Combine like terms:

-15y² = -375

Divide both sides by -15:

y² = 25

Taking the square root of both sides, we get:

y = ±5

Now substitute these values for y in either equation to find the corresponding x-values. Let's use the second equation:

x² + 2(5)² = 99
x² + 50 = 99
x² = 49
x = ±7

Therefore, the solution to the system of equations is:

(x, y) = (7, 5) and (x, y) = (-7, 5)