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One Linear Equation - One Quadratic Equation
Example
x - y = 1
x2 + y2 = 25
Method
State the linear equation
x - y = 1
Rearrange the linear equation to isolate one of the variables
x = 1 + y
x = y + 1
State the quadratic equation
x2 + y2 = 25
State the substitution
x = (y + 1)
Do the substitution to eliminate one of the variables
(y + 1)2 + y2 = 25
Do the calculation to find the two possible values of the other variable
(y + 1)(y + 1) + y2 = 25
y(y + 1) + 1(y + 1) + y2 = 25
y2 + y + y + 1 + y2 = 25
y2 + 2y + 1 + y2 = 25
2y2 + 2y + 1 = 25
2y2 + 2y + 1 - 25 = 0
2y2 + 2y - 24 = 0
y2 + y - 12 = 0
(y - 3)(y + 4) = 0
(y - 3) = 0 => y = 3
(y + 4) = 0 => y = - 4
Substitute each of these solutions in turn into the original linear equation to find the corresponding values of the other variable
State the linear equation
x - y = 1
State one of the solutions found as a substitution
y = (3)
Do the substitution
x - (3) = 1
Do the calculation to find the corresponding value of the other variable
x = 1 + 3
x = 4
State the linear equation
x - y = 1
State the other solution found as a substitution
y = (- 4)
Do the substitution
x - (- 4) = 1
Do the calculation to find the corresponding value of the other variable
x + 4 = 1
x = 1 - 4
x = - 3
x = - 3
State the two pairs of solutions of the simultaneous equations clearly
4 , 3 and -3 , - 4
We can check each of these pairs in the quadratic equation
State the quadratic equation
x2 + y2 = 25
State one pair of the substitutions
x = (4) y = (3)
Do the substitutions
(4)2 + (3)2 = 25
Do the calculation to see if left hand side equals right hand side
16 + 9 = 25
25 = 25 √
State the quadratic equation
x2 + y2 = 25
State the other pair of the substitutions
x = (- 3) y = (- 4)
Do the substitutions
(- 3)2 + (- 4)2 = 25
Do the calculation to see if left hand side equals right hand side
9 + 16 = 25
25 = 25 √
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