Problem 9 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each solution. $$ 2 a+10-3 a=9 $$

Step-by-Step Solution

Answer

The solution is \(a = 1\).

Step 1: Simplify the Equation

Start by simplifying the equation \(2a + 10 - 3a = 9\). Combine like terms \(2a\) and \(-3a\). This results in \(-a + 10 = 9\).

Step 2: Isolate the variable term

Next, we want to isolate the variable term \(-a\). To do this, subtract 10 from both sides of the equation: \(-a + 10 - 10 = 9 - 10\), resulting in \(-a = -1\).

Step 3: Solve for the variable

To solve for \(a\), multiply both sides by \(-1\) to get rid of the negative sign: \(-1 \times -a = -1 \times -1\), resulting in \(a = 1\).

Step 4: Check the solution

Substitute \(a = 1\) back into the original equation to check if it's correct: \(2(1) + 10 - 3(1) = 9\). Simplify the left side: \(2 + 10 - 3 = 9\). This simplifies to \(9 = 9\), confirming our solution is correct.

Key Concepts

Simplifying Equations

Simplifying equations is the first step in solving them. This process involves getting the equation into a simpler or more basic form. For example, if you have an equation like \(2a + 10 - 3a = 9\), simplify by seeing what can be combined or reduced on both sides of the equation.

• Look at each term in the equation. Identify and group similar terms.
• Numbers or constants can be added or subtracted from each other.
• Variables with the same base and degree, like \(2a\) and \(-3a\), can also be combined.

In our example, \(2a\) and \(-3a\) combine to become \(-a\). This step reduces the equation to \(-a + 10 = 9\). By doing this, you make it easier to isolate the variable in the next steps.

Combining Like Terms

Combining like terms is crucial when simplifying equations. Like terms in an equation are terms that have the same variable raised to the same power.Take \(2a\) and \(-3a\) from our equation, \(2a + 10 - 3a = 9\). Here, both terms contain the variable \(a\). Just like adding and subtracting regular numbers, variables are combined by performing operations on their coefficients.

• Add or subtract the coefficients of the like terms.
• Keep the variable part unchanged.

So, combining \(2a\) and \(-3a\), we perform the operation \(2 - 3\) to get \(-1\). This results in \(-a\). After combining terms, the equation looks simpler and is easier to solve: \(-a + 10 = 9\). Remember, combining like terms simplifies the equation, making each step simpler.

Isolating Variable

Isolating the variable is often viewed as the main goal of solving an equation. You want to have the variable by itself on one side of the equation to find its value.From our simplified equation \(-a + 10 = 9\), the next step is to isolate \(-a\) by moving constants away from it.

• Subtract 10 from both sides: \(-a + 10 - 10 = 9 - 10\), which becomes \(-a = -1\).
• To get \(a\) by itself, multiply both sides of the equation by \(-1\): \((-1)\times(-a) = (-1)\times (-1)\).

This multiplication gets rid of the negative sign, resulting in \(a = 1\). By isolating \(a\), you've determined its value.

Checking Solutions

Once you have a solution, it's essential to ensure it's correct. Checking solutions is a quick process that can save time and avoid errors. This involves plugging your found variable back into the original equation to see if it satisfies the equation.Taking our solution, \(a = 1\), and substituting it back into the original equation \(2a + 10 - 3a = 9\), confirms if we are right:

• Replace \(a\) with 1: \(2(1) + 10 - 3(1)\).
• Simplify: \(2 + 10 - 3\).
• The result is \(9 = 9\), which means our solution is correct.

This step is crucial. It increases confidence in your solution and ensures no errors went unnoticed through the solving process. Always check your solutions to confirm accuracy!