An Equation Has One Solution
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 tin can be hands represented graphically equally 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 as well) includes a prepare of simple procedures.
Let's empathize what are linear equations in one variable and how to solve linear equations in one variable.
What are Linear Equations in I Variable?
A linear equation in 1 variable is an equation that consists of but 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 instance, $x + ix = thirteen$ is a linear equation in i variable with the variable being $x$.
The equation above is called a linear equation in one variable considering in that location is only 1 variable ($x$) in the equation and the highest power of $x$ is 1. Such types of equations are known as linear equations in one variable.
Another examples of linear equations in i variable are $5x – i = 7$, $\frac {2x + 5}{three} = eight$, $\frac {i}{three}a + 9 = half-dozen$, $0.25x + 8 = 0$.
Maths can be really interesting for kids
Graph of Linear Equation in 1 Variable
A graph of a linear equation in i variable is always a vertical straight line crossing the $10$-centrality. The point where the line crosses the $10$-axis is chosen the root or the zero or the solution of the linear equation in one variable.
A linear equation in i variable of the grade $ax + b = 0$ crosses the $x$-axis at a indicate $(-\frac{b}{a}, 0)$. The point $(-\frac{b}{a}, 0)$ is also chosen $x$-intercept of the graph.
Allow's consider the graph of a linear equation $2x + 12 = 0$. Here, $a = 2$ and $b = 12$, therefore the graph of $2x + 12 = 0$ cantankerous $ten$-centrality at a point $\left(-\frac{12}{2}, 0\correct)$, i.e., $\left(-6, 0\right)$.
Let's consider i 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.due east., $\left(viii, 0\right)$.
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 grade of a linear equation in 1 variable is $ax + b = 0$, where $a$ is the coefficient of $x$, $x$ is the variable, and $b$ is the abiding term. In lodge to solve a linear equation in 1 variable the coefficient and the constant term should be segregated.
At present, let's run across how to segregate the coefficient and the constant term in the equation.
The two bones rules that are used in the procedure of finding the solution of a linear equation in one variable are
- If we add or subtract the aforementioned number from both sides of an equation, it still holds
- If we multiply or separate the same number into both sides of an equation, it still holds
Steps to Solve Linear Equations in Ane Variable
The steps followed to solve linear equations in 1 variable are
Step 1: Keep the variable term on ane side and constants on another side of the equation by calculation 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 respond.
Examples
Let's consider a few examples to understand the procedure of solving linear equations in 1 variable.
Ex 1: Solve $7x + ix = 0$
The showtime step is to segregate the variable and constant. Moving $9$ to the other side of the 'equal to' sign ($=$), we get
$7x = -9$
Notation:
- When a positive abiding is moved to the other side, it becomes negative on the other side
- When a negative constant is moved to the other side, information technology becomes positive on the other side
Now, isolate the coefficient from the variable by moving information technology to the other side of the 'equal to' sign ($=$).
$ten = -\frac{9}{seven}$
Annotation:
- When a coefficient that is multiplied past a variable is moved to the other side, information technology 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, information technology becomes the numerator of the fraction on the other side
Ex 2: Solve $2x – thirteen = 0$
$2x – 13 = 0$
$2x = 13$
$ten = \frac{13}{2}$
Ex 3: Solve $\frac{x}{3} + nine = 0$
$\frac{ten}{two} + 12 = 0$
$\frac{10}{2} = -12$
$10 = -12 \times 2$
$10 = -24$
Ex 4: Solve $\frac{x}{7} – 6 = 0$
$\frac{x}{7} – 6 = 0$
$\frac{x}{7} = 6$
$x = half-dozen \times 7$
$10 = 42$
Ex 5: $2 \left(ten + three \right) = 0$
$two \left(10 + 3 \right) = 0$
$10 + 3 = \frac{0}{2}$
$x + 3 = 0$
$ten = 0 – 3$
$x = – 3$
Ex 6: $\frac {7x – ii}{iii} = 0$
$7x – 2 = 0 \times 3$
$7x – 2 = 0$
$7x = 0 + 2$
$7x = 2$
$10 = \frac{2}{7}$
Solving Linear Equations in 1 Variable With Variables on Both Sides
In all the above examples of linear equations in one variable, the variable was present just on one side of the equation. Now, let's understand how to solve linear equations in 1 variable when the variable is present on both sides of the equation.
The full general process of solving linear equations in one variable with variables on both sides is the same equally that of solving linear equations in one variable with a variable on one side but.
In this case, the kickoff step is to bring the variable from the correct-mitt side to the left-hand side and so follow the steps of solving a linear equation in one variable with the variable on ane side just.
Steps to Solve Linear Equations in One Variable With Variables on Both Sides
The steps followed to solve linear equations in one variable are
Stride i: Bring the variable on the correct-hand side to the left-paw side and simplify.
Step ii: Continue the variable term on one side and constants on another side of the equation by adding or subtracting on both sides of the equation.
Stride iii: Simplify the abiding terms.
Step four: Isolate the variable on one side by multiplying or dividing information technology into both sides of the equation.
Step 5: Simplify and write the answer.
Examples
Let'southward consider a few examples to understand the process of solving linear equations in ane 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.
$9x = -ix$
$=>x = \frac{-ix}{9}$
$=>x = -1$.
Characteristics of Linear Equations in I Variable
These are some of the characteristics of linear equations in i variable.
Practice Problems
- Identify the variable, coefficient, and constant in the following equations
- $3x – v = 0$
- $2x + seven = 0$
- $4 – 5x = 0$
- $2 + 3x = 0$
- $4x – 5 = 8x$
- Solve the following equations
- $m – 5 = 0$
- $a + 3 = 0$
- $3x + seven = 0$
- $2x – 11 = 0$
- $8x – 12 = 6$
- $12x – iv = 8x$
- $15x + ii = 7x – 16$
FAQs
What is a linear equation in one variable?
Linear equation in one variable is of the class $ax + b = 0$. The linear equation in one variable are equations in which the highest degree of every term is one, in that location is one possible solution of the equation and there is only one variable present in it.
Give an example of a linear equation in one variable.
An case of a linear equation in one variable is $7m + 9 = 0$.
How many solutions does a linear equation in one variable take?
A linear equation in ane variable has a unique solution, i.e., information technology has i and simply one solution.
What is the power of a variable in a linear equation in ane variable?
The power of the variable in a one-variable linear equation is $ane$.
For case, in a linear equation in one variable $5x – 3 = 0$, the power of $x$ is $1$, since, $10$ can also exist written equally $x^{1}$.
Determination
A linear equation in one variable is an equation that consists of merely ane variable or one unknown. The standard form of a linear equation is $ax + b = 0$, where $a$ and $b$ are whatever two numbers and $ten$ is an unknown(or a variable). A linear equation in i variable has a unique (or only one) solution.
Recommended Reading
- Linear Equations in 2 Variables – Definition, Types, and Graphs
- What are Algebraic Identities(With Definition, Types & Derivations)
- What is the Meaning of Equation – Definition, Types & Examples
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- What is Algebra – Definition, Basics & Examples
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- Natural Numbers – Definition & Backdrop
- Whole Numbers – Definition & Backdrop
- What is an Integer – Definition & Properties
- Rationalize The Denominator(With Examples)
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An Equation Has One Solution,
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