Solving Simple Algebraic Equations Study Guide
Introduction to Solving Simple Algebraic Equations
Strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations.
—Ernst Mach (1838–1916) Austrian Physicist and Philosopher
In this lesson, you'll learn how to solve one-step algebraic equations.
So far, we have seen algebraic terms only in expressions. In this lesson, we will look at algebraic equations. An equation presents two expressions that are equal to each other. 3 + 6 = 9, and even 3 = 3 is an equation. An algebraic equation is an equation that includes at least one variable.
Working with algebraic expressions has taught us many of the skills we need to solve equations. The goal of solving equations is to get the variable alone on one side of the equation. If the variable is alone on one side, its value must be on the other side, which means the equation is solved.
x + 4 = 10
This equation has one variable, x. We can see that 4 is added to x, and the sum of x and 4 is equal to 10. How do we get x alone on one side of the equation? We need to get rid of that 4. We can do that by subtracting 4 from both sides of the equation. Why do we subtract 4 from both sides? The equal sign in an equation tells us that the quantities on each side of the sign have the same value. If we perform an action on one side, such as subtraction, we must perform the same action on the other side, so that the two sides of the equation stay equal. This is the most important rule when solving equations: Whatever we do to one side of an equation, we do the same to the other side of the equation.
Let's subtract 4 from both sides:
x + 4 – 4 = 10 – 4
x = 6
Our answer is x = 6. This means that if we substitute 6 for x in the equation x + 4 = 10, the equation will remain true. Some equations have more than one answer, but this equation has just one answer.
y –3 = 13
The first step in solving an equation is to find the variable. In this equation, the variable is y: y is what we must get alone on one side of the equation.
The second step in solving an equation is determining what operation or operations are needed to get the variable alone. In the previous example, we used subtraction. Why? Because a constant, 4, was added to x. We used the opposite of addition, subtraction, to get x alone. Addition and subtraction are opposite operations. Multiplication and division are also opposite operations.
The third step is to perform the operation on both sides of the equation. If we are left with the variable alone on one side of the equation, then we have our answer. If not, then we must repeat steps two and three until we have our answer.
In the equation y – 3 = 13, 3 is subtracted from y. We must use the opposite of subtraction, addition, to get y alone on one side of the equation. Add 3 to both sides of the equation:
y – 3 + 3 = 13 + 3
y = 16
When a variable in an equation has a coefficient, we must use division to get the variable alone. Remember, a coefficient and a base in a term are multiplied. Division is what we use to undo the multiplication.
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