Introduction:
Linear equations are foundational in algebra and mathematics in general. They are equations where the highest power of the variable is o ne. Thus, understanding how to solve linear equations step by step is crucial for tackling more complex problems in math and real-life scenarios. As such, this blog will guide you on how to go about the question; How Do You Solve Linear Equations Step by Step? Ensuring that by the end, you’ll be equipped to handle these problems with confidence.
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What is a Linear Equation?
A linear equation is an equation of the form ax + b = 0, where ( a ) and ( b ) are constants, and ( x ) is the variable. The goal is to find the value of ( x ) that makes the equation true. For example, in the equation ( 2x + 3 = 7 ), we need to find ( x ).
How Do You Solve Linear Equations Step by Step?
1. Simplify Both Sides of the Equation:
Firstly, before you can solve the equation, simplify both sides as much as possible. This means combining like terms and eliminating any unnecessary parentheses.
Example:
( 3(x + 2) = 2(x – 1) )
First, distribute the constants:
( 3x + 6 = 2x – 2 )
Secondly, move Variable Terms to One Side:
Thus, to isolate the variable ( x ), you need to get all the ( x )-terms on one side of the equation and the constants on the other.
Example:
( 3x + 6 = 2x – 2 )
Consequently, subtract ( 2x ) from both sides:
( 3x – 2x + 6 = -2 )
Simplify:
( x + 6 = -2 )
Thirdly, move Constant Terms to the Other Side:
Now, move all the constant terms to the opposite side of the equation.
Example:
( x + 6 = -2 )
Subtract 6 from both sides:
( x = -2 – 6 )
Simplify:
( x = -8 )
Fourthly, verify the Solution:
Notably, it is always a good practice to substitute your solution back into the original equation to ensure it satisfies the equation.
Example:
Substitute ( x = -8 ) into the original equation ( 3(x + 2) = 2(x – 1) ):
( 3(-8 + 2) = 2(-8 – 1) )
( 3(-6) = 2(-9) )
( -18 = -18 )
Since both sides are equal, ( x = -8 ) is the correct solution.
Solving Linear Equations with Fractions:
1. Eliminate Fractions by Multiplying:
Firstly, when the equation involves fractions, eliminate them by multiplying every term by the least common multiple (LCM) of the denominators.
Example:
1/2x + 1/3 = 5/6
The LCM of 2, 3, and 6 is 6. Multiply every term by 6:
( 6 {1/2}x + 6 {1/3} = 6 {5/6} )
( 3x + 2 = 5 )
2. Solve as Usual:
Then, solve the equation as you would with any linear equation.
Example:
( 3x + 2 = 5 )
Subtract 2 from both sides:
( 3x = 3 )
Divide by 3:
( x = 1 )
Special Cases in Solving Linear Equations:
1. No Solution:
Occasionally, simplifying the equation leads to a false statement, indicating there is no solution.
Example:
( 2x + 3 = 2x + 5 )
Subtract ( 2x ) from both sides:
( 3 = 5 )
This is a contradiction, so there is no solution.
2. Infinite Solutions:
Comparatively, simplifying the equation leads to a true statement, indicating there are infinitely many solutions.
Example:
( 2(x + 3) = 2x + 6 )
Distribute:
( 2x + 6 = 2x + 6 )
Subtract ( 2x ) from both sides:
( 6 = 6 )
This is always true, so there are infinitely many solutions.
How to Solve Word Problems Using Linear Equations:
1. Translate the Problem into an Equation:
Solving world problems with the application of linear equations would necessitate that one reads the problem carefully and identify what quantities the variable represents. Afterwards, Set up an equation based on the relationships described in the problem.
Example:
A train travels 100 miles in 2 hours more than it takes for a car to travel 60 miles. If the speed of the car is ( x ) mph, what is the speed of the train?
So, let the speed of the car be ( x ) mph. As such, the time taken by the car is ( 60 / x ) hours.
The time taken by the train is ( {60 / x} + 2 ) hours. Let the speed of the train be ( y ) mph, then ({100 / y} = {60 / x} + 2 ).
2. Solve the Equation:
Finally, rearrange the equation to solve for the unknown.
Example:
Cross-multiplying gives:
( 100x = y(60 + 2x) )
( 100x = 60y + 2xy )
Rearrange:
( 100x – 2xy = 60y )
( x(100 – 2y) = 60y )
( x = {60y}{100 – 2y} )
To solve this, additional information about the speed is needed, typically provided in a word problem.
Conclusion:
Conclusively, solving linear equations step by step is a crucial skill in mathematics. Thus, by understanding the process of simplifying equations, moving terms, and verifying solutions, you can confidently tackle various problems. Hence, whether dealing with simple equations or word problems, these steps provide a clear framework for finding solutions. Practice regularly, and soon you’ll master the art of solving linear equations with ease.
