Linear Equations in Two Variables: Standard Form

Standard form of a linear equation in two variables is a way to represent a straight line. It’s a concise and organized way to express the relationship between the x and y coordinates of any point that lies on that line. This form makes it easier to analyze the equation, find intercepts, and determine the slope of the line.

Formulae

The standard form of a linear equation is:

$ax + by + c = 0$

Where:

• $a$, $b$, and $c$ are real numbers.
• $a$ and $b$ are not both zero (otherwise, it wouldn’t be a linear equation in two variables).
• $a$ is the coefficient of $x$.
• $b$ is the coefficient of $y$.
• $c$ is the constant term.

Examples

Example-1 Convert the equation $y = 2x + 3$ to standard form.

Solution: Rearrange the equation to get all the terms on one side:

Subtract $2x$ and $3$ from both sides: $-2x + y – 3 = 0$. Therefore, $a=-2$, $b=1$, and $c=-3$.

Example-2 Convert the equation $3y = 6x – 9$ to standard form.

Solution: Rearrange the equation to get all the terms on one side:

Subtract $6x$ from both sides and add $9$ to both sides: $-6x + 3y + 9 = 0$. Therefore, $a=-6$, $b=3$, and $c=9$. Notice you could also divide the entire equation by -3 to get a simpler form: $2x – y – 3 = 0$.

Common mistakes by students

• Forgetting the ‘0’ on the Right Side: Students often leave out the “= 0” when writing the equation in standard form. Remember, the standard form *always* equals zero.
• Incorrectly Identifying Coefficients: Confusing the signs or misidentifying the values of $a$, $b$, and $c$ is a common error. Carefully compare the given equation with the standard form and pay attention to the signs.
• Not Clearing Fractions/Decimals: When given equations with fractions or decimals, some students may fail to convert them to integer coefficients. It’s often preferable to eliminate fractions or decimals to simplify the equation and avoid errors.

Real Life Application

Standard form can be used in real-world situations, such as modeling the cost of goods or services.

Example: Imagine you’re selling handmade crafts. Let $x$ represent the number of items you sell and $y$ represent your total revenue. If each item sells for $10, then your revenue can be represented by: $y = 10x$. To put this in standard form, subtract $y$: $-10x + y = 0$. Here, $a = -10$, $b = 1$, and $c = 0$. This illustrates a simple linear relationship between the number of items sold and the revenue earned.

Fun Fact

The standard form of a linear equation can be generalized to represent hyperplanes in higher-dimensional spaces. For instance, in 3D space, the standard form is $ax + by + cz + d = 0$, representing a plane.

Recommended YouTube Videos for Deeper Understanding

Q.1 What is the slope of the line represented by the equation $3x + 2y – 6 = 0$?

Check Solution

Ans: B

Rearrange the equation to slope-intercept form: $y = -\frac{3}{2}x + 3$. The slope is the coefficient of $x$.

Q.2 Which of the following equations represents a line parallel to $4x – y + 7 = 0$?

Check Solution

Ans: B

Parallel lines have the same slope. The given line’s slope is $4$. Rearrange options to find the one with a slope of $4$. $8x – 2y + 1 = 0$ can be written as $y = 4x + \frac{1}{2}$, so its slope is $4$.

Q.3 The line $ax + 5y + 10 = 0$ passes through the point $(1, -3)$. What is the value of $a$?

Check Solution

Ans: B

Substitute the point $(1, -3)$ into the equation: $a(1) + 5(-3) + 10 = 0$. Simplify: $a – 15 + 10 = 0$, so $a – 5 = 0$ and $a = 5$.

Q.4 What is the x-intercept of the line $2x – 3y + 12 = 0$?

Check Solution

Ans: B

The x-intercept is the point where $y=0$. Substitute $y=0$ into the equation: $2x – 3(0) + 12 = 0$. Simplify: $2x + 12 = 0$, so $2x = -12$ and $x = -6$.

Q.5 The lines $x + 2y – 4 = 0$ and $2x + ky + 8 = 0$ are perpendicular. What is the value of $k$?

Check Solution

Ans: D

The slopes of perpendicular lines multiply to -1. Slope of $x + 2y – 4 = 0$ is $-\frac{1}{2}$. The slope of $2x + ky + 8 = 0$ is $-\frac{2}{k}$. Therefore, $(-\frac{1}{2})(-\frac{2}{k}) = -1$. Simplifying $\frac{1}{k} = -1$, so $k = -1$.

Next Topic: Solutions of Linear Equations in Two Variables