Solving Linear Equations Step-by-Step

Linear equations are the foundation of algebra. They help us solve real-world problems using mathematical relationships.A linear equation is a mathematical statement where two expressions are equal and contain variables with only first-degree terms.Let's look at some examples of linear and non-linear equations. Notice how linear equations only have variables raised to the first power.When we graph a linear equation, it always forms a straight line. This is why we call it linear.Linear equations follow the principle of equality - what we do to one side, we must do to the other side to maintain balance.First-degree terms are terms where variables like x are only raised to the power of 1. Let's look at some examples.These are all first-degree terms - they contain x only to the first power.These are not first-degree terms - they contain x raised to other powers or in other forms.When solving linear equations, our first step is to combine like terms to simplify the equation.Like terms are terms that have the same variable raised to the same power. Here, 3x and 2x are like terms because they both have x to the first power.Let's understand what makes terms 'like terms'.Let's look at a more complex example with multiple like terms.First, we group all terms with x together, and all constant terms together.Then we combine the like terms: two x plus three x plus x equals six x, and five minus two equals three.It's important to recognize terms that are not like terms and cannot be combined.Let's try one more example to practice combining like terms.First, we group the terms with x and the constant terms.Then we combine four x minus x plus five x to get eight x, and two minus one to get one.When solving for a variable, we need to isolate it by dividing or multiplying both sides by the coefficient.The coefficient is the number multiplied by our variable. In this case, the coefficient is 5.To solve for x, we divide both sides by 5.This gives us x equals fourteen fifths.Which is equal to two point eight.Sometimes we need to multiply instead of divide. For example, if we have one-third x equals four.We multiply both sides by three to eliminate the fraction.This gives us x equals twelve.Let's review the steps for solving for a variable.After finding a solution, we must verify it by substituting it back into the original equation.Let's substitute x equals 2.8 into our equation: 2x plus 3 equals 11.First, we multiply 2 times 2.8 to get 5.6.Then we add 3 to get 8.6.Since both sides are equal, our solution is correct!Now let's see what happens when we check an incorrect solution.Let's substitute x equals 4 into the equation: 3x minus 4 equals 5.First multiply: 3 times 4 is 12.Then subtract 4: 12 minus 4 is 8, which does not equal 5!Since the left side doesn't equal the right side, we know this solution is incorrect.Here are some important tips to remember when checking your solutions.Always substitute back into the original equation, not a modified version. Show each step clearly. Check your arithmetic carefully. And remember, both sides must be exactly equal for the solution to be correct.