Welcome to our lesson on linear equations. A linear equation is an equation where the variable has a power of 1, there are no products of variables, and no functions of variables. Examples include equations like 2x plus 3 equals 7, 5y minus 2 equals 3y plus 8, and x divided by 2 plus 4 equals 10. The graph shows a linear equation y equals 2x minus 3, which forms a straight line. The point (2,1) lies on this line because when x equals 2, y equals 1, satisfying the equation.
Now, let's look at the steps to solve a linear equation. Step 1: Simplify both sides by removing parentheses and combining like terms. Step 2: Move all variable terms to one side of the equation by adding or subtracting. Step 3: Move all constant terms to the other side using addition or subtraction. Step 4: Isolate the variable by multiplying or dividing both sides. Let's see these steps in action with the equation 3 times open parenthesis x minus 2 close parenthesis plus 4 equals 2x plus 5. First, we distribute the 3 to get 3x minus 6 plus 4 equals 2x plus 5. Then we combine like terms to get 3x minus 2 equals 2x plus 5. Next, we subtract 2x from both sides to get x minus 2 equals 5. Finally, we add 2 to both sides, resulting in x equals 7.
Let's look at how to solve equations with fractions. When dealing with fractions, the first step is to multiply all terms by the least common denominator or LCD. This eliminates the fractions, making the equation easier to solve. Then, we solve the resulting equation using our standard steps. Finally, we always check our solution by substituting it back into the original equation. Let's solve the equation x over 3 plus 1 over 2 equals 5 over 6. First, we identify that the LCD of the denominators 3, 2, and 6 is 6. We multiply every term by 6. This gives us 6 times x over 3 plus 6 times 1 over 2 equals 6 times 5 over 6. Simplifying, we get 2x plus 3 equals 5. Subtracting 3 from both sides gives us 2x equals 2. Finally, dividing both sides by 2 gives us x equals 1. To check our solution, we substitute x equals 1 back into the original equation: 1 over 3 plus 1 over 2 equals 2 over 6 plus 3 over 6, which equals 5 over 6. Our solution is correct!
Now let's tackle equations with variables on both sides. The strategy is to move all variable terms to one side, usually the left, and all constant terms to the other side. Then combine like terms and solve for the variable. Let's solve 4x minus 7 equals 2x plus 5. First, we subtract 2x from both sides to get all variable terms on the left: 4x minus 2x equals 5 plus 7. Combining like terms, we get 2x equals 12. Finally, dividing both sides by 2 gives us x equals 6. To check our solution, we substitute x equals 6 back into the original equation: 4 times 6 minus 7 equals 24 minus 7, which is 17. And 2 times 6 plus 5 equals 12 plus 5, which is also 17. So our solution is correct! We can also visualize this graphically. The two sides of our equation represent two lines: y equals 4x minus 7 and y equals 2x plus 5. The solution is the x-coordinate of their intersection point, which is x equals 6, corresponding to y equals 17.
Let's summarize what we've learned about solving linear equations. Linear equations contain variables with a power of 1. To solve them, follow these general steps: First, simplify both sides by removing parentheses and combining like terms. Second, move all variable terms to one side of the equation. Third, move all constant terms to the other side. Fourth, isolate the variable by dividing or multiplying as needed. And finally, always check your solution by substituting it back into the original equation. For equations with fractions, multiply all terms by the least common denominator first to eliminate the fractions. Remember that the solution to a linear equation can be visualized as the intersection point of two lines on a graph. These techniques will help you solve a wide variety of linear equations that you'll encounter in algebra and beyond.