Problem 38 - Algebra 1: Concepts and Skills

Question

Find three ordered pairs that are solutions of the equation. $$ y-3 x=9 $$

Step-by-Step Solution

Answer

The three ordered pairs that are solutions of the equation are (1, 12), (2, 15), and (3, 18).

Step 1: Selecting Arbitrary Values for x

For simplicity, let us choose three arbitrary values for \(x\). Let's say \(x=1\), \(x=2\), and \(x=3\).

Step 2: Solving the Equation for y

Now, we plug these values of \(x\) into the equation \(y-3x=9\) to solve for \(y\).

Step 3: Calculating For x = 1

For \(x = 1\), the equation becomes \(y - 3(1) = 9\), which simplifies to \(y = 9 + 3 = 12\).

Step 4: Calculating For x = 2

For \(x = 2\), the equation becomes \(y - 3(2) = 9\), which simplifies to \(y = 9 + 6 = 15\).

Step 5: Calculating For x = 3

For \(x = 3\), the equation becomes \(y - 3(3) = 9\), which simplifies to \(y = 9 + 9 = 18\).

Key Concepts

Ordered Pairs

Ordered pairs are used to represent points on the Cartesian coordinate system. Each ordered pair is written as \( (x, y) \) where \( x \) is the value on the horizontal axis (also known as the x-axis), and \( y \) is the value on the vertical axis (y-axis).

In the context of solving linear equations like \( y - 3x = 9 \), each ordered pair that satisfies the equation represents a point on the line defined by that equation. For instance, if we find that \( (1, 12) \) satisfies the equation, it means that when \( x = 1 \), \( y \) is 12, and this point lies on the line.

Here’s why ordered pairs are useful:

• They help in graphing the equation on a coordinate plane.
• They can be used to identify features of a line, such as the slope and intercepts.
• They allow us to see the relationship between variables visually.

The process of identifying ordered pairs involves finding the values of \( y \) for different \( x \) values and pairing them together to form solutions.

Solving for y

Solving for \( y \) in a linear equation means isolating \( y \) on one side of the equation. For the equation \( y - 3x = 9 \), we can find \( y \) by performing simple algebraic operations.

To solve for \( y \), we need to add \( 3x \) to both sides of the equation. This yields:\[ y = 9 + 3x \]This equation expresses \( y \) in terms of \( x \), making it easier to calculate \( y \) for any given \( x \) value. By plugging different \( x \) values into this equation, we can compute the corresponding \( y \) values.

Some benefits of solving for \( y \) include:

• It simplifies the process of finding specific values of \( y \) for various \( x \) inputs.
• It transforms the original equation into the slope-intercept form, providing insights into the graph’s slope and y-intercept.
• It aids in verifying ordered pairs more swiftly, ensuring they are solutions to the equation.

Arbitrary Values for x

Choosing arbitrary values for \( x \) is a standard approach when finding ordered pairs or graphing a linear equation. Arbitrary means that you can choose almost any number for \( x \), and in this exercise, the values 1, 2, and 3 were chosen for simplicity.

By selecting these simple numbers, you can easily compute the corresponding \( y \) values using the equation. Here is why it's beneficial to choose arbitrary values for \( x \):

• It allows for straightforward calculations when the equation is in a simple form.
• Makes it easier to identify trends and relationships, such as how changes in \( x \) affect \( y \).
• Helps in building intuition about the equation’s graph and slope.

To conclude, when you pick arbitrary \( x \) values and solve for \( y \), you are building a set of points that lie on the graph of the equation. This practice is not only crucial in creating graphs but also in deepening the understanding of the relationship between variables in a linear equation.