Simultaneous equations involving a linear and a quadratic equation present a unique challenge in algebra. This section focuses on the intricacies of solving these equations, focusing on the substitution and elimination methods, and is enriched with detailed illustrative examples.

Substitution Method

The substitution method is about taking one equation, solving for one variable, and putting that solution into another equation. This is great when you can easily solve one of the equations for a variable.

Steps to Follow:

1. Pick a Variable: Look at your first equation and solve it for one variable. Like if you have $y = 2x + 3$, $y$ is ready to go.

2. Swap It In: Take that solution and replace the same variable in your second equation. If your second equation is $y^2 = 4x + 12$, switch $y$ with $2x + 3$ to get $(2x + 3)^2 = 4x + 12$.

3. Solve the New Equation: Work through the math to solve this new equation for $x$. This might mean expanding, simplifying, and using methods to solve quadratic equations.

4. Find the Other Variable: Once you have $x$, plug it back into one of the original equations to get $y$.

Elimination Method

The elimination method is about getting rid of one variable by adding or subtracting the equations from each other. This works well when you can make the coefficients (the numbers in front of the variables) match up.

Steps to Follow:

1. Match Coefficients: Adjust the equations so the same variable has the same coefficient in both. For example, if you have $3x + y = 7$ and $x^2 + y^2 = 25$, you might need to change the first equation so the $ys$ match.

2. Eliminate a Variable: Add or subtract the equations to cancel out one variable. Make sure your terms are lined up correctly before you do this.

3. Solve for One Variable: Solve the equation you get after elimination. This could be simple or more complex, depending on the equations.

4. Get the Other Variable: Take the solution from step 3 and use it in one of the original equations to find the other variable.

Examples

Example 1: Use Substitution to Solve Equations

Equations: $y = 2x + 3$ and $y^2 = 4x + 12$.

Steps:

1. Start with $y$ from the first equation.

2. Put $2x + 3$ for $y$ in the second equation: $(2x + 3)^2 = 4x + 12$.

3. Expand and simplify to $4x^2 + 8x - 3 = 0$.

4. Solve for $x$ using the quadratic formula.

5. Find $y$ by putting $x$ back into $y = 2x + 3$.

Solutions:

• First Solution: $x$ is a bit less than 0.5, and $y$ is a bit more than 4.3.
• Second Solution: $x$ is a bit less than -2.3, and $y$ is a bit less than -1.3.

Example 2: Use Elimination to Solve Equations

Equations: $3x + y = 7$ and $x^2 + y^2 = 25$.

Steps:

1. Change the first equation to $y = 7 - 3x$.

2. Replace $y$ in the second equation: $x^2 + (7 - 3x)^2 = 25$.

3. Expand and simplify to $10x^2 - 42x + 24 = 0$.

4. Solve for $x$ using the quadratic formula or factorization.

5. Get $y$ by substituting $x$ back into $y = 7 - 3x$.

Solutions:

• First Solution: $x$ is slightly less than 1, and $y$ is slightly more than 4.
• Second Solution: $x$ is significantly more than 3, and $y$ becomes negative.