Problem 71 - Algebra A Combined Function

Question

Complete each ordered pair for the given equation. $$ y=2 x+1 ;(4, \quad) $$

Step-by-Step Solution

Answer

The completed ordered pair is \((4, 9)\).

Step 1: Understand the Problem

We need to complete the ordered pair \((4, lank)\) for the equation \(y = 2x + 1\). This means we need to find the value of \(y\) when \(x = 4\).

Step 2: Substitute the x-value

Substitute \(x = 4\) into the equation \(y = 2x + 1\). This gives us:\[y = 2(4) + 1\]

Step 3: Calculate

Calculate the expression from Step 2:\[y = 2(4) + 1 = 8 + 1 = 9\]

Step 4: Complete the Ordered Pair

The ordered pair is completed by using the calculated \(y\)-value. So, the ordered pair is \((4, 9)\).

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is a simple way to display two related numbers. It often represents points on a graph, such as coordinates in the form \( (x, y) \). The first number denotes the position along the horizontal \( x \)-axis, and the second number shows the position along the vertical \( y \)-axis.
For example, in the ordered pair \( (4, 9) \), the number 4 is the \( x \)-coordinate, and 9 is the \( y \)-coordinate.

• Ordered pairs are crucial in representing solutions of equations in a two-dimensional plane.
• They provide a clear and compact form to describe points or solutions in relation to each other.

Understanding ordered pairs is essential for visualizing equations on a graph, making it easier to interpret and analyze data or geometric figures.

Linear Equations

Linear equations are algebraic expressions that create straight lines when graphed. They are usually in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept.
These equations represent a constant rate of change, meaning the relationship between the variables is consistent.

• The slope, \( m \), tells us how steep the line is and which direction it goes. A positive \( m \) indicates the line rises from left to right, while a negative \( m \) means it falls.
• The \( y \)-intercept, \( b \), is where the line crosses the \( y \)-axis, indicating the starting value when \( x = 0 \).

Linear equations are fundamental in solving real-world problems, helping us understand relationships between quantities and predict trends.

Substitution Method

The substitution method is a straightforward algebraic technique used to find the values of variables in a system of equations. It involves replacing one variable with an expression derived from another equation.
This method is particularly useful when one of the equations is solved for a single variable.

• Start by isolating one variable in one of the equations. It's often easiest to solve for the variable that is already alone on one side.
• Next, substitute this expression into the other equation, replacing the isolated variable.
• Simplify the equation and solve for the remaining variable.

Once you have solved for one variable, substitute it back into one of the original equations to find the other variable. The substitution method is a powerful tool to solve not only linear equations but also more complex systems, offering a clear step-by-step way to discover solutions.