Linear Equations in One Variable – Graph & Method of Solving

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Linear equations are widely used in many fields to find values of unknown quantities. The linear equation in one variable is the basic equation that can be easily represented graphically as a straight line. Solving a linear equation and finding the value of unknowns generally represented by a symbol (or variable) $x$ (can be represented by any other letter or symbol also) includes a set of simple procedures. 

Let’s understand what are linear equations in one variable and how to solve linear equations in one variable.

What are Linear Equations in One Variable?

A linear equation in one variable is an equation that consists of only one variable or one unknown.  The standard form of a linear equation is $ax + b = 0$, where $a$ and $b$ are any two numbers and $x$ is an unknown(or a variable). 

For example, $x + 9 = 13$ is a linear equation in one variable with the variable being $x$.

The equation above is called a linear equation in one variable because there is only one variable ($x$) in the equation and the highest power of $x$ is one. Such types of equations are known as linear equations in one variable. 

Some other examples of linear equations in one variable are $5x – 1 = 7$, $\frac {2x + 5}{3} = 8$, $\frac {1}{3}a + 9 = 6$, $0.25x + 8 = 0$.

Graph of Linear Equation in One Variable

A graph of a linear equation in one variable is always a vertical straight line crossing the $x$-axis. The point where the line crosses the $x$-axis is called the root or the zero or the solution of the linear equation in one variable.

A linear equation in one variable of the form $ax + b = 0$ crosses the $x$-axis at a point $(-\frac{b}{a}, 0)$. The point $(-\frac{b}{a}, 0)$ is also called $x$-intercept of the graph.

Let’s consider the graph of a linear equation $2x + 12 = 0$. Here, $a = 2$ and $b = 12$, therefore the graph of $2x + 12 = 0$ cross $x$-axis at a point $\left(-\frac{12}{2}, 0\right)$, i.e., $\left(-6, 0\right)$.

Let’s consider one more example of a linear equation $3x – 24 = 0$. Here, $a = 3$ and $b = -24$, therefore the graph of $3x – 24 = 0$ cross $x$-axis at a point $\left(-\left(\frac{-24}{3}\right), 0\right)$, i.e., $\left(8, 0\right)$.

You can see from the above graphs that when in a linear equation $ax + b = 0$,

• $a$ and $b$ are of the same sign, the solution is negative
• $a$ and $b$ are of opposite signs, the solution is positive

Solving Linear Equations in One Variable

The general form of a linear equation in one variable is $ax + b = 0$, where $a$ is the coefficient of $x$, $x$ is the variable, and $b$ is the constant term. In order to solve a linear equation in one variable the coefficient and the constant term should be segregated.

Now, let’s see how to segregate the coefficient and the constant term in the equation.

The two basic rules that are used in the process of finding the solution of a linear equation in one variable are 

• If we add or subtract the same number from both sides of an equation, it still holds
• If we multiply or divide the same number into both sides of an equation, it still holds

Steps to Solve Linear Equations in One Variable

The steps followed to solve linear equations in one variable are

Step 1: Keep the variable term on one side and constants on another side of the equation by adding or subtracting on both sides of the equation.

Step 2: Simplify the constant terms.

Step 3: Isolate the variable on one side by multiplying or dividing it into both sides of the equation.

Step 4: Simplify and write the answer.

Examples

Let’s consider a few examples to understand the process of solving linear equations in one variable.

Ex 1: Solve $7x + 9 = 0$

The first step is to segregate the variable and constant. Moving $9$ to the other side of the ‘equal to’ sign ($=$), we get

$7x = -9$

Note:

• When a positive constant is moved to the other side, it becomes negative on the other side
• When a negative constant is moved to the other side, it becomes positive on the other side

Now, isolate the coefficient from the variable by moving it to the other side of the ‘equal to’ sign ($=$).

$x = -\frac{9}{7}$

Note:

• When a coefficient that is multiplied by a variable is moved to the other side, it becomes the denominator of the fraction on the other side
• When a coefficient that is divided by a variable is moved to the other side, it becomes the numerator of the fraction on the other side

Ex 2: Solve $2x – 13 = 0$

$2x – 13 = 0$

$2x = 13$

$x = \frac{13}{2}$

Ex 3: Solve $\frac{x}{3} + 9 = 0$

$\frac{x}{2} + 12 = 0$

$\frac{x}{2} = -12$

$x = -12 \times 2$

$x = -24$

Ex 4: Solve $\frac{x}{7} – 6 = 0$

$\frac{x}{7} – 6 = 0$

$\frac{x}{7} = 6$

$x = 6 \times 7$

$x = 42$

Ex 5: $2 \left(x + 3 \right) = 0$

$2 \left(x + 3 \right) = 0$

$x + 3 = \frac{0}{2}$

$x + 3 = 0$

$x = 0 – 3$

$x = – 3$

Ex 6: $\frac {7x – 2}{3} = 0$

$7x – 2 = 0 \times 3$

$7x – 2 = 0$

$7x = 0 + 2$

$7x = 2$

$x = \frac{2}{7}$

Solving Linear Equations in One Variable With Variables on Both Sides

In all the above examples of linear equations in one variable, the variable was present only on one side of the equation. Now, let’s understand how to solve linear equations in one variable when the variable is present on both sides of the equation.

The general process of solving linear equations in one variable with variables on both sides is the same as that of solving linear equations in one variable with a variable on one side only.

In this case, the first step is to bring the variable from the right-hand side to the left-hand side and then follow the steps of solving a linear equation in one variable with the variable on one side only.

Steps to Solve Linear Equations in One Variable With Variables on Both Sides

The steps followed to solve linear equations in one variable are

Step 1: Bring the variable on the right-hand side to the left-hand side and simplify.

Step 2: Keep the variable term on one side and constants on another side of the equation by adding or subtracting on both sides of the equation.

Step 3: Simplify the constant terms.

Step 4: Isolate the variable on one side by multiplying or dividing it into both sides of the equation.

Step 5: Simplify and write the answer.

Examples

Let’s consider a few examples to understand the process of solving linear equations in one variable with variables on both sides.

.

Ex 1: Solve $12x + 9 = 3x$.

$12x + 9 = 3x$

Bring the variable on RHS to LHS.

$12x – 3x + 9 = 0$

$9x + 9 = 0$

Now, follow the steps of solving a linear equation in one variable.

$9x = -9$

$=>x = \frac{-9}{9}$

$=>x = -1$.

Characteristics of Linear Equations in One Variable

These are some of the characteristics of linear equations in one variable.

Practice Problems

1. Identify the variable, coefficient, and constant in the following equations
   • $3x – 5 = 0$
   • $2x + 7 = 0$
   • $4 – 5x = 0$
   • $2 + 3x = 0$
   • $4x – 5 = 8x$
2. Solve the following equations
   • $m – 5 = 0$
   • $a + 3 = 0$
   • $3x + 7 = 0$
   • $2x – 11 = 0$
   • $8x – 12 = 6$
   • $12x – 4 = 8x$
   • $15x + 2 = 7x – 16$

FAQs

Conclusion

A linear equation in one variable is an equation that consists of only one variable or one unknown.  The standard form of a linear equation is $ax + b = 0$, where $a$ and $b$ are any two numbers and $x$ is an unknown(or a variable). A linear equation in one variable has a unique (or only one) solution.

Recommended Reading

• Linear Equations in Two Variables – Definition, Types, and Graphs
• What are Algebraic Identities(With Definition, Types & Derivations)
• What is the Meaning of Equation – Definition, Types & Examples
• Division of Algebraic Expressions(With Methods & Examples)
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• Natural Numbers – Definition & Properties
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• What is an Integer – Definition & Properties
• Rationalize The Denominator(With Examples)
• Multiplication of Irrational Numbers(With Examples)