Find the value of a variable that satisfies an equation
Determine whether an equation can be solved with a single answer, cannot be solved at all, or has an infinite number of possible solutions
Algebraic Expressions
The Main Idea Algebraic expressions are mathematical statements that combine numbers or constants, variables (letters that represent unknown numbers), and operations such as addition, subtraction, multiplication, and division. They serve as the foundational language for algebra and allow us to represent and solve complex mathematical problems. An algebraic expression can be as simple as a single variable “[latex]x[/latex]“, or as complex as a multi-term expression like “[latex]3x^2 - 2x + 5[/latex]“.
The parts of the expression separated by addition or subtraction are called terms, each of which can be a combination of numbers (coefficients) and variables raised to a power. To evaluate an algebraic expression, you substitute specific numerical values for the variables and perform the indicated operations. For instance, if we substitute [latex]x=2[/latex] into the expression “[latex]3x^2 - 2x + 5[/latex]“, we would get [latex]3*(2)^2 - 2*2 + 5 = 12 - 4 + 5 = 13[/latex].
The Main Idea Simplifying algebraic expressions is the process of making an expression as concise as possible without changing its value. It involves several steps, often relying on mathematical properties and operations to combine like terms and eliminate unnecessary parts.
One fundamental process is the combination of like terms, which are terms in the expression that have the same variables and exponents. For example, in the expression [latex]3x + 2y - 5x + y[/latex], you can combine the [latex]3x[/latex] and [latex]-5x[/latex] to get [latex]-2x[/latex], and the [latex]2y[/latex] and [latex]y[/latex] to get [latex]3y[/latex].
Thus, the expression simplifies to [latex]-2x + 3y[/latex]. The distributive property is also frequently used in simplification. It allows us to remove parentheses in expressions like [latex]3(x + 2)[/latex], which becomes [latex]3x + 6[/latex] after distribution.
A rectangle with length [latex]L[/latex] and width [latex]W[/latex] has a perimeter [latex]P[/latex] given by [latex]P=L+W+L+W[/latex]. Simplify this expression.
The Main Idea Multi-step expressions refer to algebraic expressions that require multiple operations to simplify or solve. The process involves a combination of mathematical techniques including, but not limited to, applying the order of operations, combining like terms, and using properties of real numbers such as distributive, associative, and commutative properties.
Solve [latex]3y+2=11[/latex]
Subtract [latex]2[/latex] from both sides of the equation to get the term with the variable by itself.
In the two videos that follow, we show examples of how to solve an absolute value equation that requires you to isolate the absolute value first using mathematical operations.
Watch the following video for a demonstration of how to solve a multi-step equation containing fractions by using the least common denominator to clear the fractions first.
The Main Idea An equation with no solutions, also known as an inconsistent equation, is an equation that has no value for its variable that will make it a true statement. These equations often result from manipulating an original equation in a way that produces a mathematical impossibility. When you attempt to solve these types of equations, you end up with an expression that contradicts itself. For instance, an equation like [latex]2x + 3 = 2x + 5[/latex] simplifies to [latex]3 = 5[/latex], a statement that is always false regardless of what value [latex]x[/latex] may have. This is an example of an equation with no solutions.
Try solving these equations. How many steps do you need to take before you can tell whether the equation has no solution or one solution?
a) Solve [latex]8y=3(y+4)+y[/latex]
First, distribute the [latex]3[/latex] into the parentheses on the right-hand side.
[latex]8y=3y+12+y[/latex]
Next, begin combining like terms.
[latex]8y=4y+12[/latex]
Now move the variable terms to one side. Moving the [latex]4y[/latex] will help avoid a negative sign.
Now, take a moment to ponder this equation. It says that [latex]2x-10[/latex] is equal to [latex]2x+7[/latex]. Can some number times two minus [latex]10[/latex] be equal to that same number times two plus seven? Pretend [latex]x=3[/latex]. Is it true that [latex]2\left(3\right)-10=-4[/latex] is equal to [latex]2\left(3\right)+7=13[/latex]. NO! We do not even really need to continue solving the equation, but we can just to be thorough. Add [latex]10[/latex] to both sides.
We know that [latex]0\text{ and }17[/latex] are not equal, so there is no number that [latex]x[/latex] could be to make this equation true. This false statement implies there are no solutions to this equation, or [latex]DNE[/latex] (does not exist) for short.
Again, we have a result where an absolute value is negative! There is no solution to this equation, or [latex]DNE[/latex].
Equations with Many Solutions
The Main Idea An equation with many solutions, also known as an identity, is an equation that is true for all values of its variable. These equations often arise when both sides of an equation reduce to the same expression or if the equation is a universally true statement. For example, consider the equation [latex]3x + 2 = 3x + 2[/latex]. No matter what value [latex]x[/latex] is, the left and right side of the equation will always be equal. Hence, this equation is an identity and has an infinite number of solutions since all real numbers can satisfy the equation. When you attempt to solve these types of equations, you end up with an expression that is always true, such as [latex]5 = 5[/latex]. An equation of this sort indicates that the original equation is true for any value of the variable.
Solve for [latex]x[/latex].
[latex]3\left(2x-5\right)=6x-15[/latex]
Distribute the [latex]3[/latex] through the parentheses on the left-hand side.
Wait! This looks just like the previous example. You have the same expression on both sides of an equal sign. No matter what number you choose for [latex]x[/latex], you will have a true statement. We can finish the algebra: