Answer:
the solution to the system of equations is:
x = -32
y = -21
Explanation:
To solve the system of equations using algebraic methods, you can use the method of either substitution or elimination. In this example, I'll use the elimination method to solve for x and y. Here are the equations:
3x - 5y = 9
-4x + 2y = 2
We can start by manipulating the equations so that when added or subtracted, one of the variables cancels out. Let's start by multiplying both sides of the second equation by 5 to make the coefficients of y in both equations cancel out:
3x - 5y = 9
(-4x + 2y) * 5 = 2 * 5
This gives us:
3x - 5y = 9
-20x + 10y = 10
Now, we can add these two equations to eliminate the variable y:
(3x - 5y) + (-20x + 10y) = 9 + 10
Now, simplify:
(3x - 20x) + (-5y + 10y) = 19
-17x + 5y = 19
Now, we have a new equation:
-17x + 5y = 19
Now, we have a system of two equations with two variables:
3x - 5y = 9
-17x + 5y = 19
Next, you can either use the elimination or substitution method to solve for x and y. Let's use the elimination method again to eliminate y:
Multiply the first equation by 5 to make the coefficients of y in both equations cancel out:
5(3x - 5y) = 5(9)
This gives us:
5(3x) - 5(5y) = 45
15x - 25y = 45
Now, add this equation to the fourth equation:
(15x - 25y) + (-17x + 5y) = 45 + 19
Now, simplify:
(15x - 17x) + (-25y + 5y) = 64
-2x = 64
Now, divide both sides by -2 to solve for x:
-2x / -2 = 64 / -2
x = -32
Now that we have found the value of x, you can substitute it back into one of the original equations to solve for y. Using the first equation:
3x - 5y = 9
3(-32) - 5y = 9
-96 - 5y = 9
Now, add 96 to both sides:
-5y = 9 + 96
-5y = 105
Finally, divide both sides by -5 to solve for y:
-5y / -5 = 105 / -5
y = -21