24. Solve the system of equations algebraically. Show all of your steps.
Y = x^2 + 2x and y = 3x + 20. Btw this is a workpad problem. I just need to be put in the right direction, on how to do it.
set the equations equal (both equal y)
3x+20=x^2+2x
x^2-x-20=0
(x-5)(x+4)=0
solutions at x=5, and x=-4
Now put those solutions into either equation, and you have a x,y set.
y=x^2+2x
(-4,8) and (5,35) are the two solutions.
Aww. Thanks Guys! I really appreiciate it.
Wait how did bobpusrsly get x^2-x-20=0
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To get x^2-x-20=0, you can set the two given equations equal to each other because they are both equal to y.
So, y = x^2 + 2x becomes:
y = 3x + 20 (equation 1)
x^2 + 2x = 3x + 20 (substituting y with 3x + 20 from equation 1)
x^2 - x - 20 = 0 (subtracting 3x and 20 from both sides)
This is a quadratic equation in standard form (ax^2 + bx + c = 0) where a=1, b=-1, and c=-20. To solve for x, you can factor or use the quadratic formula.
To solve the system of equations algebraically, we need to set up the equations, which you have already provided:
Equation 1: Y = x^2 + 2x
Equation 2: Y = 3x + 20
Now, to find the solution, we need to find the value of x that satisfies both equations. Since both equations are equal to Y, we can set them equal to each other:
x^2 + 2x = 3x + 20
To solve this quadratic equation, rearrange it and set it equal to zero:
x^2 + 2x - 3x - 20 = 0
Combine like terms:
x^2 - x - 20 = 0
Now, we need to factorize or use the quadratic formula to solve for x. In this case, let's use factoring:
(x - 5)(x + 4) = 0
Setting each factor equal to zero, we find two possible solutions for x:
x - 5 = 0 or x + 4 = 0
Solving for x in each equation:
x = 5 or x = -4
Now that we have the x-values, we can substitute them back into either equation to find the corresponding y-values.
Substituting x = 5 into Equation 1:
Y = (5)^2 + 2(5) = 25 + 10 = 35
So, when x = 5, y = 35.
Substituting x = -4 into Equation 1:
Y = (-4)^2 + 2(-4) = 16 - 8 = 8
So, when x = -4, y = 8.
Hence, the solution to the system of equations is x = 5, y = 35 and x = -4, y = 8.