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How to Solve Simultaneous Equations with a Quadratic and Linear Equation
Mathematics
Grade 10 (Junior High School)
Question Content
Solve the following simultaneous equations: $y = x^2$; $11x + 2y = 6$
Correct Answer
$x=-6, y=36$ and $x=\\frac{1}{2}, y=\\frac{1}{4}$
Detailed Solution Steps
1
Step 1: Substitute $y = x^2$ into the linear equation $11x + 2y = 6$. This gives $11x + 2(x^2) = 6$.
2
Step 2: Rearrange the equation into standard quadratic form $ax^2+bx+c=0$. Rearranging gives $2x^2 + 11x - 6 = 0$.
3
Step 3: Solve the quadratic equation. We can factor it as $(2x - 1)(x + 6) = 0$, or use the quadratic formula $x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$ where $a=2$, $b=11$, $c=-6$.
4
Step 4: Calculate the $x$-values: From $(2x - 1)=0$, we get $x=\\frac{1}{2}$; from $(x + 6)=0$, we get $x=-6$.
5
Step 5: Substitute each $x$-value back into $y=x^2$ to find the corresponding $y$-values. For $x=-6$, $y=(-6)^2=36$; for $x=\\frac{1}{2}$, $y=(\\frac{1}{2})^2=\\frac{1}{4}$.
Knowledge Points Involved
1
Simultaneous Equations (Substitution Method)
This method is used to solve systems of equations where one equation is substituted into the other to eliminate one variable, reducing the system to a single-variable equation. It is especially useful when one equation is already solved for a variable, like $y=x^2$ here.
2
Quadratic Equations (Standard Form and Factoring)
A quadratic equation has the standard form $ax^2+bx+c=0$. Factoring rewrites the quadratic as a product of two linear binomials, allowing us to use the zero product property (if $ab=0$, then $a=0$ or $b=0$) to find solutions. For unfactorable quadratics, the quadratic formula is used.
3
Quadratic Formula
The formula $x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$ gives the solutions to any quadratic equation $ax^2+bx+c=0$. The discriminant $b^2-4ac$ tells us the number and type of solutions: positive for two real solutions, zero for one real solution, negative for two complex solutions.
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