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Solving equations involves finding the value of the unknown variables in the given equation. The condition that the two expressions are equal is satisfied by the value of the variable. Solving a linear equation in one variable results in a unique solution, solving a linear equation involving two variables gives two results. Solving a quadratic equation gives two roots. There are many methods and procedures followed in solving an equation. Let us discuss the techniques in solving an equation one by one, in detail.
What is the Meaning of Solving Equations?
Solving equations is computing the value of the unknown variable still balancing the equation on both sides. An equation is a condition on a variable such that two expressions in the variable have equal value. The value of the variable for which the equation is satisfied is said to be the solution of the equation. An equation remains the same if the LHS and the RHS are interchanged. The variable for which the value is to be found is isolated and the solution is obtained. Solving an equation depends on what type of equation that we are dealing with. The equations can be linear equations, quadratic equations, rational equations, or radical equations.
Steps in Solving an Equation
The aim of solving an equation is to find the value of the variable that satisfies the condition of the equation true. To isolate the variable, the following operations are performed still balancing the equation on both sides. By doing so LHS remains equal to RHS, and eventually, the balance remains undisturbed throughout.
- Addition property of equality: Add the same number to both the sides. If a = b, then a + c = b + c
- Subtraction property of equality: Subtract the same number from both sides. If a = b, then a - c = b - c
- Multiplication property of equality: Multiply the same number on both sides. If a = b, then ac = bc
- Division property of equality: Divide by the same number on both sides. If a = b, then a/c = b/c (where c ≠ 0)
After performing this systematic balancing method of solving an equation by a series of identical arithmetical operations on both sides of the equation, we separate the variable on one of the sides and the ultimate step is the solution of the equation.
Solving Equations of One Variable
A linear equation of one variable is of the form ax + b = 0, where a, b, c are real numbers. The following steps are followed while solving an equation that is linear.
- Remove the parenthesis and use the distributive property if required.
- Simplify both sides of the equation by combining like terms.
- If there are fractions, multiply both sides of the equation by the LCD (Least common denominator) of all the fractions.
- If there are decimals, multiply both sides of the equation by the lowest power of 10 to convert them into whole numbers.
- Bring the variable terms to one side of the equation and the constant terms to the other side using the addition and subtraction properties of equality.
- Make the coefficient of the variable as 1, using the multiplication or division properties of equality.
- isolate the variable and get the solution.
Consider this example: 3(x + 4) = 24 + x
We simplify the LHS using the distributive property.
3x + 12 = 24 + x
Group the like terms together using the transposing method. This becomes 3x - x = 24-12
Simplify further ⇒ 2x = 12
Use the division property of equality, 2x/2 = 12/2
isolate the variable x. x = 6 is the solution of the equation.
Use any one of the following techniques to simplify the linear equation and solve for the unknown variable. The trial and error method, balancing method and the transposing method are used to isolate the variable.
Solving an Equation by Trial And Error Method
Consider 12x = 60. To find x, we intuitively try to find that 12 times what number is 60. We find that 5 is the required number. Solving equations by trial and error method is not always easy.
Solving an Equation by Balancing Method
We need to isolate the variable x for solving an equation. Let us use the separation of variables method or the balancing method to solve it. Consider an equation 2x + 3 = 17.
We first eliminate 3 in the first step. To keep the balance while solving the equation, we subtract 3 from either side of the equation.
Thus 2x + 3 - 3 = 17 - 3
We have 2x = 14
Now to isolate x, we divide by 2 on both sides. (Division property of equality)
2x/2 = 14/2
x = 7
Thus, we isolate the variable using the properties of equality while solving an equation in the balancing method.
Solving an Equation by Transposing Method
While solving an equation, we change the sides of the numbers. This process is called transposing. While transposing a number, we change its sign or reverse the operation. Consider 5y + 2 = 22.
We need to find y, so isolate it. Hence we transpose the number 2 to the other side. The equation becomes,
5y = 22-2
5y = 20
Now taking 5 to the other side, we reverse the operation of multiplication to division. y = 20/5 = 4
Solving an Equation That is Quadratic
There are equations that yield more than one solution. Quadratic polynomials are of degree two and the zeroes of a quadratic polynomial represent the quadratic equation.
Consider (x+3) (x+2)= 0. This is quadratic in nature. We just equate each of the expressions in the LHS to 0.
Either x+3 = 0 or x+2 =0.
We arrive at x = -3 and x = -2.
A quadratic equation is of the form ax2 + bx + c = 0. Solving an equation that is quadratic, results in two roots: α and β.
Steps involved in solving a quadratic equation are:
- By Completing The Squares Method
- By Factorization Method
- By Formula Method
By Completing The Squares Method
Solving an equation of quadratic type by completing the squares method is quite easy as we apply our knowledge of algebraic identity: (a+b)2
- Write the equation in the standard form ax2 + bx + c = 0.
- Divide both the sides of the equation by a.
- Move the constant term to the other side
- Add the square of one-half of the coefficient of x on both sides.
- Complete the left-hand side as a square and simplify the right-hand side.
- Take the square root on both sides and solve for x.
For more information about solving equations (quadratic) by completing the squares, click here.
By Factorization Method
Solving an equation of quadratic type using the factorization method, follow the steps discussed here. Write the given equation in the standard form and by splitting the middle terms, factorize the equation. Rewrite the equation obtained as a product of two linear factors. Equate each linear factor to zero and solve for x. Consider 2x2 + 19x + 30 =0. This is of the standard form ax2 + bx + c = 0.
Split the middle term in such a way that the product of the terms should equal the product of the coefficient of x2 and c and the sum of the terms should be b. Here the product of the terms should be 60 and the sum should be 19. Thus, split 19x as 4x and 15x (as the sum of 4 and 15 is 19 and their product is 60).
2x2 + 4x + 15x + 30 = 0
Take the common factor out of the first two terms and the common factors out of the last two terms.
2x(x + 2) + 15(x + 2) = 0
Factoring (x+2) again, we get
(x + 2)(2x + 15) = 0
x = -2 and x = -15/2
Solving an equation that is quadratic involves such steps while splitting the middle terms on factorization.
By Formula Method
Solving an equation of quadratic type using the formula
x = [-b ± √[(b2 -4ac)]/2a helps us find the roots of the quadratic equation ax2 + bx + c = 0. Plugging in the values of a, b, and c in the formula, we arrive at the solution.
Consider the example: 9x2 -12 x + 4 = 0
a= 9, b = -12 and c = 4
x = [-b ± √[(b2 -4ac)]/2a
= [12 ± √[((-12)2 -4×9×4)] / (2 × 9)
= [12 ± √(144 - 144)] / 18
= (12 ± 0)/18
x = 12/18 = 2/3
Solving an Equation That is Rational
An equation with at least one polynomial expression in its denominator is known as the rational equation. Solving an equation that is rational involves the following steps. Reduce the fractions to a common denominator and then solve the equation of the numerators.
Consider x/(x-1) = 5/3
On cross-multiplication, we get
3x = 5(x-1)
3x = 5x - 5
3x - 5x = - 5
-2x = -5
x = 5/2
Solving an Equation That is Radical
An equation in which the variable is under a radical is termed the radical equation. Solving an equation that is a radical involves a few steps. Express the given radical equation in terms of the index of the radical and balance the equation. Solve for the variable.
Consider √(x+1) = 4
Now square both the sides to balance it. [ √(x+1)]2 = 42
(x+1) = 16
Thus x = 16-1 =15
Important Notes on Solving Equations:
- Solving an equation is finding the value of the variable in the equation.
- The solution of an equation satisfies the condition of the given equation.
- Solving an equation of linear type can be also done graphically.
- If the right side part of an equation is zero, then for solving equation, just graph the left side of the equation and the x-intercept(s) of the graph would be the solution(s).
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Examples of Solving an Equation
Example 1. Use the balancing method of solving equations: (x-2) / 5 - (x-4) / 2 = 2
The given equation is (x-2) / 5 - (x-4) / 2 = 2.
Solving an equation that is rational involves the following steps.
Simplify the LHS. Take the LCD of the denominators. LCD is 10.
[2(x-2) -5(x-4)]/10 = 2
Use distributive property and simplify the numerator.
We get [2x- 4 -5x+20]/10 = 2
Use the multiplication property of equality to get rid of the denominator.
10 × [2x- 4 -5x+20]/10 = 10 × 2
Simplifying we get - 3x + 16 = 20
isolate the term with the variable using the addition property of equality
-3x + 16 - 16 = 20 - 16
-3x = 4
isolate the variable using the division property of equality
-3x/3 = 4/3
x = -4/3
Answer: Thus solving (x-2)/ 5 - (x-4)/2 = 2, x = -4/3
Example 2. Use the transposing method of solving an equation 0.4(a+10)= 2 - 0.6a
Solving equations that are linear involving decimals involves the following steps.
Given 0.4(a+10)= 2 - 0.6a
0.4 a + 0.4 × 10 = 2- 0.6a
0.4 a + 4 = 2- 0.6a
0.4 a + 0.6a = 2-4
1 a = -2
a = -2
Answer: The solution is a = -2
Example 3. What is the value of p on solving an equation: 4 (p - 3) - (p - 5) = 4?
Given: 4 (p - 3) - (p - 5) = 4
Let us use the transposing method in solving equations.
4p - 12 - p + 5 = 4 (distributive property)
3p - 7 = 4
3p = 4 + 7
3p = 11
p = 11/3
Answer: The value of p = 11/3
Practice Questions on Solving an Equation
FAQs on Solving Equations
What is Solving an Equation?
Solving an equation is finding the value of the unknown variables in the given equation. The process of solving an equation depends on the type of the equation.
What are The Steps in Solving Equations?
Identify the type of equation: linear, quadratic, logarithmic, exponential, radical or rational.
- Remove the brackets, if any in the given equation. Apply the distributive property.
- Add the same number to both the sides
- Subtract the same number from both the sides
- Multiply the same number on both the sides
- Divide by the same number on both sides.
What are The Golden Rule in Solving an Equation?
The type of the equation is identified. If it is a linear equation, separating the variables method or transposing method is used. If it is a quadratic equation, completing the squares, splitting the middle terms using factorization is used or by formula method.
How Do You use 3 Steps in Solving an Equation?
The 3 steps in solving an equation are to
- remove the brackets, if any using the distributive property,
- simplify the equation by adding or subtracting the like terms,
- isolating the variable and solving it.
How Do You Solve Linear Equations?
While solving an equation that is linear, we isolate the variable whose value is to be found. We either use transposing method or the balancing method.
How Do You Solve Quadratic Equations?
While solving an equation that is quadratic, we write the equation in the standard form ax2 + bx + c = 0, and then solve using the formula method or factorization method or completing the squares method.
How Do You Solve Radical Equations?
While solving an equation that is radical, we remove the radical sign, by raising both the sides of the equation to the index of the radical, isolate the variable and solve for x.
How Do You Solve Rational Equations?
While solving an equation that is rational, we simplify the expression on each side of the equation, cross multiply, combine the like terms and then isolate the variable to solve for x.