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Reducing equations is a method for solving a complex equation and bringing it in a simpler form.
- A linear equation is a first-order equation.
- Not all equations are in the form of linear equations.
- They can be solved by converting them to linear equations using mathematical operations such as cross-multiplication.
- After converting these nonlinear equations to linear form, they can be solved, and the value of the variable calculated simply.
| Table of Content |
Key Terms: Linear Equation, Non-Linear Equations, Simple Equations, Reducing Equations, Cross Multiplication method, Distributive law
Reducing Equation to Simpler Form
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Certain mathematical equations are not written in the form of linear equations, but they can be converted into linear equations by performing specific mathematical operations.
- Reducing an equation to its simplest form is known as reduction.
- It is the process of simplifying and reducing nonlinear equations to linear ones.
- These equations can be solved and the unknown value is determined after converting to linear form.
- Reducing such equations simplifies calculations and saves the time needed to solve them.
- Performing the simplification involves a variety of mathematical methods, which vary depending on the needs of a certain equation.
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Linear and Nonlinear Equations
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An equation is a statement of equivalence between two algebraic expressions that contain constants and variables. The degree and variable in the equations determine whether they are linear or nonlinear.
Linear Equations
A linear equation is one that has a term with the maximum degree of one. Alternatively, we can say that a linear equation with only one variable is known as a linear equation in one variable.
- A linear equation value, when plotted on a graph, provides a straight line.
- A linear equation has the following general form: ax + b = c, where a, b, and c are constants and x is a variable.
Nonlinear Equations
A nonlinear equation is one that has a term with a maximum degree of two or more.
- When you graph nonlinear equations, you'll see that they show as curved lines.
- When using unknown variables, the discrepancies between the equation's results are unpredictable, the equation is nonlinear.
- Nonlinear equations can take a variety of forms, ranging from simple lines to complex pictures.
- Nonlinear equations don't have powers of one in them.
Steps for Reducing Equation to Simpler Form
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The steps to simplify the equation are:
- If the provided solution is non-linear, it cannot be computed directly. As a result, we must first use the Cross Multiplication approach to simplify the following problem.
- Both parts of the equation are cross-multiplied, meaning the denominator on one side is multiplied by the numerator on the other.
- To release the brackets, use the distributive law.
- Place all of the variables on one side of the problem (LHS) and the constants on the other (RHS).
- Solve the remainder of the problem as a linear equation with one variable.
Example of Reducing Equation into Simpler Form
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To understand the Simplification better, Look at the below example:
Solve: x – 1 / x + 2 = 1 / 6
Step 1
Since the problem is non-linear, it cannot be calculated directly. As a result, we must first use the Cross Multiplication approach to simplify the following problem.
x – 1 / x + 2 = 1 / 6
The denominator on both sides is multiplied by the numerator from the other side in the cross-multiplication approach.
Step 2
The equation can be expressed as follows after cross-multiplication:
6 (x – 1) = 1 (x + 2)
Step 3
Now, using distributive law, open the parenthesis.
6x – 6 = x + 2
Step 4
Bring all of the variables to one side (LHS), and all of the constants to the other side (RHS).
6x – x = 2 + 6
⇒ 5x = 8
Step 5
By multiplying both sides by 5, you can get the answer.
x = 8/5
Methods to Simplify Equation
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The methods to simplify the equations are given below:
- Multiply each term in the parenthesis by the term outside of the parentheses.
- This basically refers to spreading the outer term into the inner terms.
- Multiply the expression outside of the parentheses by the parentheses' first term.
- Multiply the result by the second term. When there are more than two terms, continue to distribute them until none remain.
- Maintain the parenthesis for any operation (plus or minus).
- Combine phrases that are similar. Must combine like terms before you can solve the equation.
- Add all of the numerical terms together. Merge any variable components separately.
- Arrange the terms such that the variables are on one side of the equals sign and the constants (numbers only) are on the other to simplify the equation.
- Spread a negative number along with its corresponding negative sign.
- If you're multiplying a phrase or term within parentheses with a negative value, make sure you disperse the negative to each term inside the parenthesis.
Things to Remember
- In a variety of professional situations, nonlinear equations may be used more frequently than linear equations.
- These equations can help in managing projects and forecasting.
- When the presence of different variables is entered into a linear equation, the variance between the outputs is always the same.
- When several unknown variables are entered into a nonlinear equation, however, the disparities between the different outputs vary from time to time.
- The simplest type is the integer in its smallest corresponding fraction.
- Certain arithmetic equations are not in the set of linear equations, but they can be made into linear equations by applying particular algebraic calculations.
- These problems can be derived and the value of the unknown is simply estimated when they have been reduced to linear form.
- A statistical equation that shows the connection between two variables on both edges of the 'equal to' sign is referred to as a simple calculation.
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Sample Questions
Ques. What is a reducible equation? (1 Mark)
Ans. A reducible equation is one that can be simplified by "reducing" its terms to get an equivalent equation that is easier to solve.
Ques. What is a linear equation? (2 Marks)
Ans. A linear equation is an algebraic equation of the form y = mx + b that has just a constant and a first-order (linear) component, where m is the slope and b is the y-intercept.
Ques. What is a non-linear equation? (2 Marks)
Ans. A nonlinear equation is characterized as having a maximum degree of two or more. A linear equation provides a straight line on a graph. A nonlinear equation forms a curve on a graph.
Ques. What is the Cross Multiplication Method? (3 Marks)
Ans. When solving linear equations with two variables, the cross-multiplication approach is utilized. The technique of cross-multiplication is the simplest and most straightforward way of solving linear equations in two variables. When there are two variables in a linear equation, this approach is commonly used.
Cross-multiplication is a method for finding the solution to two-variable linear equations. It turns out to be the quickest way to solve a set of linear equations. For a given set of two-variable linear equations:
Ques. What is Distributive law? (3 Marks)
Ans. The responsiveness of the function value (exit value) to a change in its argument is measured by the derivative of a value of a real variable (input value). Calculus uses derivatives as a fundamental tool. The velocity of a moving item, for example, is the derivative of its position with respect to time: it quantifies how quickly the object's position changes as time passes.
When a derivative of a particular function exists at a given input value, it is the gradient of the tangent line to the function's graph at that point. The tangent line is the function's closest linear estimate near that input quantity.
Ques. What is an example of a linear equation? (2 Marks)
Ans. To calculate variable expenses, you can use a linear equation. For example, if you hire a professional painter who charges $100 per day and $0.20 per square foot painted, you can use a linear equation to calculate how much you will spend for the painting service. In this case, x is the number of square feet that need to be painted, and the linear equation to utilize is:
0.20x + 100 = y
Ques. Solve: x – 1 = x/3 + 3/4 (3 Marks)
Ans. Calculate the L.C.M. of the third and fourth denominators, which is 12.
= (x – 1) = (4x – 9)/12
= 12x −12 = 4x + 9.
= 12x − 4x = 9 + 12.
= 8x = 21.
= x = 21/8 [Divide both the sides by 8]
Ques. What is a Reduction Method? (2 Marks)
Ans. Rewriting a statement into a simpler form is referred to as reduction. "Reducing a fraction," for instance, is the practice of rewriting a fraction into one with the lowest whole-number denominator feasible (while maintaining the numerator as a whole integer).
Reducing a radical refers to rewriting a radical (or "root") phrase with the shortest possible whole integer under the radical symbol. Denesting radicals is the process of reducing the amount of radicals that appear beneath other radicals in a statement.
Ques. What is a substitution method? (5 Marks)
Ans. Three phases are involved in the substitution method:
- For one of the variables, calculate one equation.
- Evaluate the other equation by substituting (plugging in) this expression.
- To determine the corresponding variable, substitute the value into the original system.
So, while solving systems by replacement, we'll need to keep an eye out for cases like this. We'll get an invalid answer if they're parallel and don't cross.
Ques. Solve the linear equation: (3x-4)/2 – (2x-1)/3 = (1/2)+x. (5 Marks)
Ans. Given,
(1/2)+x = (3x-4)/2 – (2x-1)/3
- For the left- and right-hand words, use the LCM.
[3(3x-4)-2(2x-1)]
(1+2x)/2 = /2.3
- To simplify the equation, remove the brackets.
(9x-12-4x+2)/6 = (1+2x)/2.
(5x-10)/6 = (1+2x)/2.
- By multiplying the above equation by two, we get
6(1+2x) = 2(5x – 10).
6 + 12x = 10x – 20.
-2x is equal to 26.
- As a result, x = -13
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