2.4 Use a General Strategy to Solve Linear Equations
3 min read•june 24, 2024
Linear equations are the building blocks of algebra. They help us solve real-world problems by turning words into math. We'll learn how to simplify, isolate variables, and find solutions step-by-step.
Understanding different types of linear equations is key. We'll explore with specific solutions, that are always true, and with . This knowledge is crucial for tackling more complex math problems.
Solving Linear Equations
Step-by-step linear equation solving
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Top images from around the web for Step-by-step linear equation solving
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Simplify each side of the equation separately
Combine like terms on each side of the equation (e.g. 3x+2x=5x)
Distribute if necessary (e.g. 2(3x−1)=6x−2)
Use addition or subtraction to get all terms on one side and all terms on the other side
Add or subtract the same value from both sides to maintain equality (e.g. if 2x+3=7, then 2x+3−3=7−3)
Combine like terms after adding or subtracting (2x=4 in the previous example)
Isolate the variable term by dividing both sides by its
Divide both sides by the same non-zero value to maintain equality (e.g. if 2x=4, then 22x=24)
Simplify the fraction if possible (x=2 in the previous example)
Check the solution by substituting the value back into the original equation (e.g. if x=2, then 2(2)+3=7 is true)
Types of linear equations
Conditional equations
Have a solution that makes the equation true
Solving the equation yields a specific value for the variable (x=2 for 2x+3=7)
Identities
True for all values of the variable
Solving results in a true statement like 0=0 (e.g. 2(x+1)=2x+2)
Equivalent expressions on both sides of the equals sign
Contradictions
False for all values of the variable
Solving results in a false statement like 0=1 (e.g. 2x+1=2x+2)
Inconsistent equations with no solution
Understanding Linear Equations
A is an that forms a straight line when graphed
Variables are symbols (usually letters) that represent unknown quantities in an equation
Constants are fixed numerical values in an equation that do not change
involves manipulating the equation to isolate the variable and find its value
Applying Linear Equations
Real-world applications of solutions
Identify the unknown quantity and assign a variable to represent it (e.g. let x = the number of apples)
Write an equation that models the situation using the given information
Translate verbal phrases into mathematical expressions (e.g. "twice the number of apples" becomes 2x)
Use appropriate units for the variable and constants (e.g. if x represents apples, the equation should not include units of length)
Solve the equation using the step-by-step approach
Interpret the solution in the context of the problem
Check if the solution makes sense in the given context (e.g. a negative number of apples is not realistic)
State the solution using appropriate units (e.g. "The solution is 5 apples.")
Verify the solution by substituting it back into the original word problem to ensure it satisfies the given conditions