The question aims to find the **standard form** of an **algebraic equation. The** question is based on the concepts of **algebraic equations**, particularly **linear equations** with **two variables. Linear equations** are algebraic equations with variables only having an **exponent** of **one.** These equations represent a **linear straight line** as shown in Figure 1. The equation of the line is given as:

\[ Ax + By = C \]

Here **A, B, and C** are constants, and **x and** **y** are **two variables.** If we solve this equation for variable y, then **A/B** will represent the **slope** of the equation, and **C/B** will give us the **y-intercept** of the **line** represented by this equation.

## Expert Answer

The given **algebraic linear** equation is:

\[ y = 2x\ -\ 9 \]

Figure 1 below shows the **graph** of the equation for $0 \leq x \leq 5$.

Figure 1 shows the **graph** of the given equation, which has a **slope of 2**, and the **y-intercept** is **-9,** as shown in the Figure above.

The **standard form** of the equation is given as:

\[ Ax + By = C \]

To make the given **linear equation** in **standard form,** we can perform the following **operations.**

\[ y = 2x\ -\ 9 \]

**Step 1: Subtract** **y** from both sides.

\[ y\ -\ y = 2x\ -\ 9\ -\ y \]

\[ 0 = 2x\ -\ 9\ -\ y \]

**Step 2: Add** **9** on both sides.

\[ 0 + 9 = 2x\ -\ 9\ -\ y + 9 \]

\[ 9 = 2x\ -\ y \]

Rearranging the equation to represent in **standard form.**

\[ 2x\ -\ y = 9 \]

When this equation is used to **plot** the **graph,** we will get the same **line** shown above in Figure 1, as these two equations are exactly the **same.**

## Numerical Result

The **standard form** of the given equation **y = 2x – 9** is calculated to be:

\[ 2x\ -\ y = 9 \]

## Example

How do you write the **algebraic equation** **y = x – 6** in **standard form?**

\[ y = x\ -\ 6 \]

Figure 2 below shows the **graph** of the **equation** for $0 \leq x \leq 5$.

The given equation has a **slope of 1**, as can be observed from the graph, and the **y-intercept is -6.**

The **standard form** of the equation is given as:

\[ Ax + By = C \]

To make the given **linear equation** in standard form, we can perform the following **operations.**

\[ y = x\ -\ 6 \]

**Step 1: ** **Subtract y from both sides.**

\[ y\ -\ y = x\ -\ 6\ -\ y \]

\[ 0 = x\ -\ 6\ -\ y \]

**Step 2:** **Add 6 on both sides.**

\[ 0 + 6 = 2x\ -\ 6\ -\ y + 6 \]

\[ 6 = x\ -\ y \]

Rearranging the equation to represent in **standard form.**

\[ x\ -\ y = 6 \]

When this equation is used to **plot** the **graph,** we will get the **same line** shown above in Figure 2, as these two equations are **exactly** the **same.**

*Images/Mathematical drawings are created with Geogebra.*