Problem 43 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(35-46,\) determine which, if any, of the ordered pairs listed satisfy the given equation. $$y=x^{2}-3 x-4 ; \quad(-2,8),(1,-6),(2,8)$$

Step-by-Step Solution

Answer

(1, -6) satisfies the given equation.

Step 1 - Understand the Given Equation

The equation provided is given by \[ y = x^2 - 3x - 4 \]We need to determine which of the given ordered pairs satisfy this equation.

Step 2 - Test the First Ordered Pair (-2, 8)

Substitute \(x = -2\) and \(y = 8\) into the equation: \[ 8 = (-2)^2 - 3(-2) - 4 \]Calculate the right-hand side: \[ 8 = 4 + 6 - 4 \]\[ 8 = 6 \]This pair does not satisfy the equation.

Step 3 - Test the Second Ordered Pair (1, -6)

Substitute \(x = 1\) and \(y = -6\) into the equation: \[ -6 = (1)^2 - 3(1) - 4 \]Calculate the right-hand side: \[ -6 = 1 - 3 - 4 \]\[ -6 = -6 \]This pair satisfies the equation.

Step 4 - Test the Third Ordered Pair (2, 8)

Substitute \(x = 2\) and \(y = 8\) into the equation: \[ 8 = (2)^2 - 3(2) - 4 \]Calculate the right-hand side: \[ 8 = 4 - 6 - 4 \]\[ 8 = -6 \]This pair does not satisfy the equation.

Key Concepts

Quadratic Equations

In algebra, a quadratic equation is any equation that can be re-arranged in the standard form \( ax^2 + bx + c = 0 \).The one here is slightly different since it's written as \( y = x^2 - 3x - 4 \). This means it's not set to zero, but rather, shows a relationship between two variables, \(x\) and \(y\).
Here's a quick look at the general behavior of quadratic equations:

• The highest power of \(x\) is 2, making it a second-degree polynomial.
• Graphically, it represents a parabola, which can either open upwards or downwards.
• The coefficient of \(x^2\) determines the direction of the parabola.

Understanding these basics helps you grasp why you might substitute pair values into the equation to see if they satisfy the relationship.

Ordered Pairs

An ordered pair is a set of numbers written in a specific order, usually \((x, y)\).
They are used to represent coordinates on a Cartesian plane and indicate the location of points.

• In our problem the pairs given are \((-2,8)\), \((1,-6)\), and \((2,8)\).
• The first value in each pair is the \(x\)-coordinate, and the second is the \(y\)-coordinate.

Using these pairs, we substitute each value of \(x\) into the quadratic equation and see if the corresponding \(y\) value aligns with the outcome.

Substitution Method

The substitution method involves replacing a variable with its given value to solve equations.
For quadratic equations, this means plugging in the \(x\) and \(y\) values from ordered pairs into the equation.

• Step 1: Take the \(x\) value from the pair and substitute it into the equation.
• Step 2: Calculate the right-hand side to see if it equals the given \(y\) value.

Example: For the pair \((1, -6)\), substitute \(x = 1\) and \(y = -6\) into our equation:
\( -6 = (1)^2 - 3(1) - 4 \)
Calculate to find:
\( -6 = 1 - 3 - 4 \)
Since both sides match, the pair satisfies the equation.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operations like addition and subtraction.
In our quadratic equation, \( y = x^2 - 3x - 4 \), the expression \( x^2 - 3x - 4 \) is algebraic.

• Algebraic expressions form the foundation for solving equations.
• They can be manipulated using various algebraic rules to find values of variables.

Understanding how to work with these expressions is key to solving more complex equations. Always simplify and perform operations step-by-step.
For example, applying this systematically for each ordered pair lets you check if they satisfy the given quadratic equation.