Problem 70 - Algebra A Combined Function

Question

Complete each ordered pair for the given equation. $$ y=-1.3 x ;(6, \quad) $$

Step-by-Step Solution

Answer

(6, -7.8)

Step 1: Understand the Equation

The given equation is \( y = -1.3x \). This equation represents a straight line with a slope of \(-1.3\). You are asked to find the value of \( y \) when \( x = 6 \).

Step 2: Plug in the Value of x

Substitute the value of \( x = 6 \) into the equation \( y = -1.3x \). This gives us \( y = -1.3 \times 6 \).

Step 3: Calculate the Value of y

Multiply \(-1.3\) by \(6\) to find \( y \). The calculation is: \( y = -1.3 \times 6 = -7.8 \).

Step 4: Complete the Ordered Pair

Now that we have found \( y = -7.8 \), the ordered pair is complete. Thus, the ordered pair is \( (6, -7.8) \).

Key Concepts

Ordered Pairs

An ordered pair is essentially a way to represent a point on a coordinate plane using two values. The first value, usually denoted as \(x\), represents the horizontal placement of the point, while the second value, \(y\), signifies the vertical position. This pair is written as \((x, y)\). When you see a point like (6, -7.8), it means this specific spot is 6 units over on the x-axis and -7.8 units along the y-axis.
Understanding ordered pairs is crucial, as they help in visualizing where points lie relative to each other on a graph. They are the backbone of plotting data and interpreting graphs, which are central to many fields like physics, engineering, and economics.
In the context of equations, for instance, when you have an equation like \(y = -1.3x\), ordered pairs can be used to find points that satisfy the equation, helping you plot the line that represents the equation on a graph.

Slope

The slope of a line indicates how steep it is, and it's a crucial concept in understanding linear equations. In the equation \(y = -1.3x\), the number \(-1.3\) is the slope. It tells us how much \(y\) changes for every one-unit increase in \(x\).

• If the slope is positive, the line rises as it moves from left to right.
• If the slope is negative, like our example, the line falls as it moves to the right.
• A larger slope value means a steeper line.

When dealing with linear equations, recognizing the slope can quickly help you determine how the line behaves without even graphing it. The slope of \(-1.3\) suggests that for every step you take along the x-axis, the line drops 1.3 units downwards.
This concept is especially useful in real-world situations, such as calculating gradients in physical terrains or understanding rates of change in financial trends.

Substitution Method

The substitution method is a straightforward way to solve equations, especially useful for finding ordered pairs in linear equations. Here's how it works:
Once you have an equation like \(y = -1.3x\), and you know one value like \(x = 6\), you "substitute" this value into the equation. This means you'll replace \(x\) with 6 in the equation.
So, substituting \(x = 6\) into \(y = -1.3x\) turns it into \(y = -1.3 \times 6\). By performing the multiplication, you determine \(y = -7.8\).

• Choose the equation and known variable.
• Substitute the known variable into the equation.
• Solve for the unknown variable.

The substitution method is powerful because it simplifies the equation and makes finding unknown values straightforward. It's widely used in linear algebra and other higher-level math areas when dealing with systems of equations.