Problem 8 - Precalculus Mathematics for Calculus

Question

Use the substitution method to find all solutions of the system of equations. $$\left\\{\begin{aligned} x^{2}-y &=1 \\ 2 x^{2}+3 y &=17 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solutions are (2, 3) and (-2, 3).

Step 1: Solve the First Equation for y

The first equation is \( x^2 - y = 1 \). Rearrange this equation to solve for \( y \): \[ y = x^2 - 1\]

Step 2: Substitute Expression for y into the Second Equation

Substitute \( y = x^2 - 1 \) into the second equation: \[ 2x^2 + 3(x^2 - 1) = 17\]This results in one equation with one variable.

Step 3: Simplify and Solve for x

Simplify the equation from Step 2: \[ 2x^2 + 3x^2 - 3 = 17 \]Combine like terms: \[ 5x^2 - 3 = 17\]Add 3 to both sides: \[ 5x^2 = 20\]Divide by 5: \[ x^2 = 4\]Take the square root of both sides: \[ x = 2 \text{ or } x = -2\]

Step 4: Solve for y Using Values of x

Substitute \( x = 2 \) and \( x = -2 \) back into \( y = x^2 - 1 \) to find \( y \):For \( x = 2 \):\[y = 2^2 - 1 = 4 - 1 = 3\]For \( x = -2 \):\[y = (-2)^2 - 1 = 4 - 1 = 3\]Thus, \( y = 3 \) for both values of \( x \).

Step 5: Write the Solutions as Ordered Pairs

The solutions of the system of equations are the ordered pairs that satisfy both equations:1. \( (2, 3) \)2. \( (-2, 3) \)

Key Concepts

System of equations

A system of equations is where two or more equations are considered together because they share common variables. In our case, we have a system of two equations:

• \( x^2 - y = 1 \)
• \( 2x^2 + 3y = 17 \)

These equations intertwine through the variables \( x \) and \( y \). In most cases, solving a system of equations means finding sets of values for \( x \) and \( y \) that satisfy all of the equations at once. This means we're searching for points where these equations "meet" or intersect in a geometrical perspective. Generally, systems can be linear or non-linear; ours is non-linear because of the \( x^2 \) terms. Understanding how these systems work is crucial in both algebra and calculus, as it helps us find where conditions overlap or cross over.

Solving equations

Solving equations involves finding the values of the variables that make the equation true. With the substitution method, which is often handy for both linear and non-linear equations, the key steps are:

• First, we solve one of the equations for a variable. In our example, we solved \( x^2 - y = 1 \) for \( y \), getting \( y = x^2 - 1 \).
• Next, we substituted this expression for \( y \) into the second equation, leading to \( 2x^2 + 3(x^2 - 1) = 17 \).
• This reduces the problem to one equation with one variable, \( x \), which is simpler to handle.
• Finally, we solved for \( x \) to find \( x = 2 \) or \( x = -2 \).

Once these solutions for \( x \) are found, they are used to find corresponding \( y \) values. It requires being systematic and methodical, often involving simplification, substitution, and sometimes factoring or using the quadratic formula.

Ordered pairs

Ordering pairs express solutions that satisfy systems of equations and describe coordinates on a plane, ensuring both equations are fulfilled. For our system:

• Once we found \( x = 2 \) and \( x = -2 \), we used \( y = x^2 - 1 \) to find \( y = 3 \) for both cases.
• This gives us two ordered pairs: \((2, 3)\) and \((-2, 3)\).

Each pair \((x, y)\) indicates that when \( x \) is 2 or -2, \( y \) becomes 3, satisfying both initial equations. Ordered pairs are particularly helpful because they offer precise solutions that can be plotted on a coordinate system. In essence, ordered pairs give us specific locations where our equations intersect, showing concrete solutions for systems of equations.