Problem 8 - Algebra and Trigonometry

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 it to solve for \(y\):\[ y = x^2 - 1 \]

Step 2: Substitute y in the Second Equation

Substitute \(y = x^2 - 1\) into the second equation \(2x^2 + 3y = 17\):\[ 2x^2 + 3(x^2 - 1) = 17 \]

Step 3: Simplify and Solve for x

Simplify the equation:\[ 2x^2 + 3x^2 - 3 = 17 \]\[ 5x^2 - 3 = 17 \]Add 3 to both sides:\[ 5x^2 = 20 \]Divide by 5:\[ x^2 = 4 \]Solve for \(x\):\[ x = 2 \quad \text{or} \quad x = -2 \]

Step 4: Find Corresponding Values of y

For \(x = 2\):\[ y = (2)^2 - 1 = 4 - 1 = 3 \]For \(x = -2\):\[ y = (-2)^2 - 1 = 4 - 1 = 3 \]

Step 5: Write the Solution

The solutions of the system are \((2, 3)\) and \((-2, 3)\).

Key Concepts

System of Equations

A system of equations is a set of two or more equations with the same variables. The solutions to these systems are the values of the variables that satisfy all equations simultaneously.
In our exercise, we have a system with two equations:

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

This system involves the variables \( x \) and \( y \). The goal is to find pair(s) of \( x \) and \( y \) values that work for both equations at the same time.
There are different methods to solve systems of equations, including:

• Graphical method
• Substitution method
• Elimination method

In this exercise, we used the substitution method which is particularly useful when one equation is easily solvable for one variable.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is about finding the unknown or putting real-life variables into equations and then solving them.
In our exercise, algebra helps us manipulate and solve the equations:

• We identify the unknowns, \( x \) and \( y \), in the given equations.
• Algebraic operations like addition, subtraction, multiplication, and division are used to simplify and solve these equations.
• Rearranging equations to isolate a variable is a key step. For instance, solving \( x^2 - y = 1 \) for \( y \) gives \( y = x^2 - 1 \).

By using algebraic techniques, we can transform complex equations into simpler forms that make finding solutions easier.

Solving Quadratic Equations

Quadratic equations are second-order polynomial equations in a single variable having the general form \( ax^2 + bx + c = 0 \). In the given system of equations:

• Equation 1 \( x^2 - y = 1 \) rearranges to express \( y \) in terms of \( x \).
• When substituted, Equation 2 becomes \( 5x^2 = 20 \), a quadratic equation.

To solve quadratic equations, you can factor, use the quadratic formula, or complete the square.
Here, the equation simplifies such that:

• Divide by 5 to isolate \( x^2 \) as \( x^2 = 4 \).
• Solve for \( x \) by taking the square root, providing \( x = 2 \) or \( x = -2 \).
• These solutions for \( x \) are further substituted back to find the corresponding \( y \) values.

Thus, it is clear how solving the quadratic equation helps find confirmed values for \( x \) that satisfy the system of equations.