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Important Information for section 8.4 System of Nonlinear Equations (two variables)
A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form Ax+By+C=0Ax+By+C=0. Any equation that cannot be written in this form in nonlinear. The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.
Methods
-Substitution
-Elimination
For Example #1 I'm going to show you Substitution.
Methods
-Substitution
-Elimination
System of equations: x² - 4y² = -7
3x² + y² = 31
For Example #2 I'm going to show you Elimination.
Q:
The jeopardy board has 30 questions on it let x be the amount of rows and let y be the amount of columns on the jeopardy board. johnny gilbert made his own jeopardy board and it has 57 questions on it, he put 3 times the number of rows and 7 times the number of columns on the normal jeopardy board. Find X and Y using the subsition method.
step 1: write out your system of equations.
x⋅y= 30
3x + 7y= 57
Step 1: eliminate x or y (I'm going to do x)
(x² - 4y² = -7) -3 3x² + y² = 31
(-3x² +21y² = 21) + (3x² + y² = 31) = 13y² = 52
step 2: find x or y
step 3: substitute
x=30/y -> 3(30/y)+7y=57 -> 90/y+7y=57 -> y(90/y+7y=57) -> 90+7y²=57y -> -57y+7y²+90=0
Step 2= find y
(13y² = 52) ÷ 13 -> y²= 4 -> y= + or - 2
Step 4: factor
(7y-15) (y-6)
Step 3= use y to find x
y=2 -> x²-4(2)²=-7
x²-4(2)²=-7 -> x²-16(+16)=-7(+16) -> x²=9 -> x=+or-3 (3,2)(-3,2)
y=-2 -> x² - 4(-2)² = -7
x²-4(-2)²=-7 -> x²-16(+16)=-7(+16) -> x²=9 -> x=+or-3 (3,-2)(-3,-2)
step 5: find x with y
y= 15/7 x=30/(15/7) x=14.018 ( 15/7 , 14.02)
y=6 x= 30/6 x=5 (6,5)
Step 4= Answers
( 3, 2 ) ( -3, 2 )
( 3, -2 ) ( -3, -2 )