équation

Once you know what something IS, then you know what you can and can’t DO with it. You might be familiar with the old adage “Once you have a hammer, every problem looks like a nail.” We’d like you to consider the reverse; once you recognize that you’re trying to use a nail, you know you should reach for your hammer! An algebraic expression consists of numbers and variables joined together with mathematical operations (such as addition, subtraction, multiplication, division, exponentiation). Remember, what we’re looking for are “numbers and variables joined together with math”, and we want to make sure we don’t have any equal signs. That makes options a and d examples of algebraic expressions. Examples b and c aren’t algebraic expressions, because they each include an equal sign! Write two examples of algebraic expressions, and two examples of things that aren’t algebraic expressions. When working with an algebraic expression, we never want to change its value. What does this mean? Well, let’s work with the super sophisticated expression of 2 to investigate. In the end, then, we’re saying that $ frac{4x}{5y}$ is the same as $ frac{8x}{10y}$, and the same as $ frac{68x}{85y}$. The new versions are just a bit fancier than the original! We’ve focused a lot on addition and multiplication; what about subtraction and division? If we are careful, we can think about subtraction and division as being special forms of addition and multiplication, respectively. The parentheses are mostly here to help us visually sort out the addition followed by a negative symbol. Now, we can rearrange. When faced with an algebraic expression, we can also do a whole host of things that usually fall under the category of “simplification”. Depending on our goals, simplification might mean distribution, combining like terms, factoring, and more. Notice that the distributive property only applies in this specific case: when we want to multiply by an addition (or subtraction) expression. Let’s take a detour to make sure we understand when distribution does and doesn’t apply. Two algebraic terms are like terms if they have the exact same combination of variables raised to the exact same powers. Determine which of the following expressions represent terms. Then, determine which are like terms. Notice, we added the coefficients of the like terms, but did nothing to change the variables. To combine like terms, add or subtract the coefficients, but do not change the variables in any way. Now we have some distribution and eventually some like terms to combine. To ensure that the negatives don’t mess us up, though, let’s change that $- 3$ into adding the opposite: 3. Our friends, fractions! We’ll review some fraction basics while using the order of operations. Again, we change the subtraction to help keep track of the signs. Now we turn our sights to equations. Equations are a lot like expressions, with one key difference: they have an equal sign! A mathematical statement in which we write that one algebraic expression equals another. Equations are written in the structure: One of our main tasks when faced with an equation is solving for a value of the variable(s) which makes the equation a true statement. When we simplify algebraic expressions, we’ll write each step on a new line, with the equal sign at the beginning, to distinguish from when we’re simplifying or evaluating versus when we’re solving an equation. Let’s dig into what we’re doing here. First of all, while we’re checking our potential solution, we use a $? =$ instead of an $=$. We’re not claiming that the two sides are equal, we’re checking whether they are indeed equal. Notice also that we’re working with each side independently, simplifying each expression using the order of operations. How can we find potential solutions to test, though? That’s a really big topic, which we’ll revisit time and again throughout this text. For now, we’ll review a few of the main concepts for solv