Problem 67 - Algebra A Combined Function

Question

Complete each ordered pair for the given equation. See Section \(3.1 .\) \(y=7 x+3 ;(4, \quad)\)

Step-by-Step Solution

Answer

The ordered pair is \((4,31)\).

Step 1: Identify the Point and Equation

We have the equation \( y = 7x + 3 \) and an ordered pair \((4, \_ )\). This means when \( x = 4 \), we need to find \( y \).

Step 2: Substitute \( x \) into the Equation

Substitute \( x = 4 \) into the equation \( y = 7x + 3 \). This gives us \( y = 7(4) + 3 \).

Step 3: Solve for \( y \)

Calculate \( y = 7(4) + 3 \). First, multiply: \( 7 \times 4 = 28 \). Then add \( 3 \): \( 28 + 3 = 31 \).

Step 4: Complete the Ordered Pair

The value of \( y \) when \( x = 4 \) is 31. Thus, the complete ordered pair is \((4, 31)\).

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics, especially when working with equations and graphs. An ordered pair consists of two elements. The first element is typically the "x" value, and the second element is the "y" value. Together, they are written in parentheses as \((x, y)\). This pair corresponds to a point on a graph.

For example, if you have the ordered pair \((4, 31)\), this tells you that at the point where \(x = 4\), the \(y\) value is 31. This is very useful for graphing equations, as it provides a specific location on the coordinate plane.

• The first number always refers to the horizontal distance from the origin (x-axis).
• The second number refers to the vertical distance (y-axis).

Ordered pairs are critical in expressing the solutions of linear equations and other mathematical functions.

Knowing how to correctly write and interpret ordered pairs is vital for understanding graphs and analyzing mathematical relationships.

Substitution Method

The substitution method is a powerful algebraic technique used to find the value of one variable using another variable's value. It involves "substituting" or "replacing" one variable in an equation with its known value.

In this context, when given the equation \(y = 7x + 3\) and an ordered pair \((4, \_ )\), the substitution method involves replacing \(x\) with 4 since it is the given value from the pair. This leads to the following calculation:

• Replace \(x\) in the equation: \(y = 7(4) + 3\).
• Solve for \(y\) by performing the arithmetic operations to find that \(y = 31\).

This method is used extensively in algebra to make solving equations much simpler by reducing the number of unknowns.

By mastering the substitution method, you can effectively tackle a wide range of mathematical problems where one variable's value is known and affects the result of another variable.

Solving Linear Equations

Solving linear equations is a core skill in algebra that allows us to find unknown values and understand relationships between variables. Linear equations typically appear in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

To solve a linear equation, one often needs to find the value of \(y\) when \(x\) is given, or vice versa. The solution process involves straightforward arithmetic. Consider the equation \(y = 7x + 3\) and the ordered pair \((4, \_ )\):

• Substitute \(x\) with 4: This converts the equation into \(y = 7(4) + 3\).
• Perform multiplication: \(7 \times 4 = 28\).
• Add the constant: \(28 + 3 = 31\).
• Thus, when \(x = 4\), \(y = 31\), completing the ordered pair as \((4, 31)\).

The process is systematic, making it easy to understand and apply.

By practicing how to solve linear equations, you're developing an essential skill that is widely applicable in various fields, such as engineering, economics, and everyday problem solving.