Problem 37 - Algebra A Combined Function

Question

Complete each ordered pair so that it is a solution of the given linear equation. See Example 5. $$ x-4 y=4 ;(\quad,-2),(4, \quad) $$

Step-by-Step Solution

Answer

Completed pairs: \((-4, -2)\) and \((4, 0)\).

Step 1: Solve for y in the first ordered pair

We have the equation \(x - 4y = 4\) and the ordered pair \((\_, -2)\). To find the value of \(x\), substitute \(-2\) for \(y\) in the equation: \[ x - 4(-2) = 4 \] which simplifies to \(x + 8 = 4\). Solve for \(x\):\[ x = 4 - 8 = -4 \] Therefore, the completed ordered pair is \((-4, -2)\).

Step 2: Solve for y in the second ordered pair

The second ordered pair given is \((4, \_)\). Substitute \(4\) for \(x\) in the equation \(x - 4y = 4\): \[ 4 - 4y = 4 \]. Simplify and solve for \(y\): \[ -4y = 0 \]. Therefore, \(y = 0\). Hence, the completed ordered pair is \((4, 0)\).

Key Concepts

Understanding Ordered Pairs

Ordered pairs are simply pairs of numbers that can represent points on a graph in a coordinate system. Each pair, typically written as \(x, y\), tells you the 'address' of a point in the two-dimensional space. This concept plays a fundamental role in identifying solutions to linear equations.

• The first element, \(x\), represents the horizontal position.
• The second element, \(y\), represents the vertical position.

When you are given an equation and asked to find corresponding ordered pairs, you are essentially finding points that lie on the line represented by the equation. In the exercise above, the equation \(x - 4y = 4\) can be associated with various such points. By plugging one of the values (either \(x\) or \(y\)) into the equation, you can determine the counterpart in the pair. This is key to graphing lines and understanding their relationships.

Solving for Variables

Solving for variables is the process of finding unknown values in equations that make them true. It involves manipulating the equation to isolate the variable on one side. Let’s take a closer look at how this can be done:1. **Substituting Known Values:** Start by substituting the given value from the ordered pair into the equation.
For instance, if we have \(x - 4y = 4\) and need to find \(x\) for a given \(y = -2\), we substitute \(-2\) for \(y\). 2. **Simplifying:** Next, simplify the equation so that all terms involving the variable you are solving for are on one side.3. **Isolating the Variable:** Solve the remaining linear equation by performing inverse operations to isolate the variable.
In the first step of the example, by substituting \(y = -2\) and simplifying, we solve \(x + 8 = 4\) to find \(x\).This method is useful not only for completing ordered pairs but also in a wide range of mathematical problems that require solving equations.

The Substitution Method

The substitution method is an important technique used to solve equations and complete ordered pairs. It involves replacing a variable in one equation with an equivalent expression from another equation, making it easier to solve for one variable at a time.In the exercise, substitution is used to determine the missing values in the ordered pairs for a given linear equation. Here’s how it works:- **Start with an Equation and a Known Value:** Take the equation \(x - 4y = 4\) and a known value, such as \(x\) or \(y\) from the ordered pair.- **Replace the Variable:** Substitute the known value into the equation to find the missing component of the ordered pair. For example, substituting \(-2\) for \(y\) to solve for \(x\).- **Solve the Resulting Equation:** With the substituted values, simplify and solve the equation to find the other variable.Using this method simplifies the process of solving complex equations by breaking them down into easier, manageable steps. It reinforces the relationship between algebraic expressions and their graphical representation.