Problem 11 - Algebra 1: Concepts and Skills

Question

Solve the equation if possible. Determine whether the equation has one solution, no solution, or is an identity. $$ 12-5 a=-2 a-9 $$

Step-by-Step Solution

Answer

The equation has a unique solution, \( a = 7 \).

Step 1: Simplify both sides of the equation

First, let's simplify the equation by collecting like terms on both sides. We can add \( 5a \) to both sides of the equation to cancel the \( -5a \) term on the left side and the \( -2a \) term on the right. Doing so results in the equation \( 12 = 3a - 9 \).

Step 2: Solve for the variable

Secondly, to isolate \( a \) on one side, let's add \( 9 \) to both sides of our equation resulting in \( 21 = 3a \). Finally, to solve for \( a \), we can divide both sides by \( 3 \), yielding \( a = 7 \).

Step 3: Check the type of solution

Lastly, in order to determine whether our equation has a unique solution, no solution, or is an identity, we replace \( a \) with \( 7 \) in our original equation. The left side \( 12 - 5(7) \) equals \( -23 \) and the right side \( -2(7) - 9 \) also equals \( -23 \). Since both expressions are equal, the equation has a unique solution.

Key Concepts

Linear Equations

Linear equations are one of the first types of algebraic equations you learn about in math. They are called "linear" because they represent straight lines when graphed on a coordinate plane. A typical linear equation can be written in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. When solving linear equations, we manipulate these elements to find the value of \( x \) that makes the equation true. Linear equations are the stepping stones to understanding more complex mathematical concepts such as systems of equations and calculus.

• Simplicity: Linear equations generally involve just addition, subtraction, multiplication, or division of terms.
• Graphing: They produce a straight line on a graph, which makes them easy to work with visually.

Unique Solution

A unique solution in equations means there is exactly one value for the variable that satisfies the equation. In the context of the equation given: \( 12 - 5a = -2a - 9 \), the unique solution was found to be \( a = 7 \). Unique solutions are indicated by the fact the variable can be isolated, and a single value satisfies both sides of the equation.

• Single Intersection: When graphed, a linear equation with a unique solution will intersect the \( x\)-axis at only one point.
• Verifying: You can substitute the solution back into the original equation to verify that both sides are equal, confirming the solution is indeed unique.

Isolate Variable

To solve an equation, one crucial step is to isolate the variable. This involves arranging the equation so the variable stands alone on one side. This makes it easier to determine its value. In our example, after simplifying, the equation becomes \( 12 = 3a - 9 \). Isolating \( a \) involves two main actions: first, adding 9 to both sides to get \( 21 = 3a \), and second, dividing both sides by 3 to find \( a = 7 \). This process of isolation is essential when solving for unknowns.

• Balance: Always perform the same operation on both sides to maintain the equality of the equation.
• Goal: The end goal is to have the variable on one side and the numbers on the other.

Like Terms

"Like terms" are terms that contain the same variables raised to the same power. Recognizing and working with like terms is important in solving linear equations because it simplifies the equation, making it easier to solve. In the equation \( 12 - 5a = -2a - 9 \), the terms \(-5a\) and \(-2a\) are considered like terms because they both involve the variable \( a \).

• Simplifying: Combining like terms on either side of an equation often results in fewer terms, simplifying the equation.
• Identifying: Terms with numerical coefficients but identical variables can be directly added or subtracted.