Problem 24 - Algebra 2 and Trigonometry

Question

Solve for the variable and check. Each solution is an integer. \((2 x+1)+(4-3 x)=10\)

Step-by-Step Solution

Answer

The solution is \(x = -5\).

Step 1: Combine Like Terms

Simplify the expression on the left side of the equation by combining the like terms. Start with the variable terms: \((2x - 3x) + (1 + 4) = -x + 5\). This simplifies the equation to \(-x + 5 = 10\).

Step 2: Isolate the Variable

To isolate \(x\), subtract 5 from both sides of the equation:\(-x + 5 - 5 = 10 - 5\).This reduces to \(-x = 5\).

Step 3: Solve for the Positive Variable

Multiply both sides of the equation \(-x = 5\) by -1 to solve for \(x\):\((-1)(-x) = (5)(-1)\).This results in \(x = -5\).

Step 4: Check the Solution

Substitute \(x = -5\) back into the original equation to ensure it is correct:\[ (2(-5) + 1) + (4 - 3(-5)) = 10 \].Simplifying within the parentheses:\[ (-10 + 1) + (4 + 15) = 10 \].So it simplifies to:\[ -9 + 19 = 10 \].The left side equals the right side, 10, so \(x = -5\) is the correct solution.

Key Concepts

Combining Like Terms

When solving algebraic equations, one of the foundational steps is combining like terms. This involves simplifying expressions by merging terms that have the same variable part. Consider the equation \[(2x + 1) + (4 - 3x) = 10\]To begin, focus on the variable terms. In this case, these are \(2x\) and \(-3x\). Combining these gives us \(2x - 3x\), which simplifies to \(-x\).
Next, calculate the constant terms: \(1\) and \(4\), which sum up to \(5\). Together, this combining results in the equation \[-x + 5 = 10\]This simplification helps set the stage for further steps in solving for the variable.

Isolating Variables

Once you've combined like terms, the next step in solving algebraic equations is isolating the variable. The goal here is to get the variable by itself on one side of the equation. Using our simplified equation \[-x + 5 = 10\]we start by eliminating the constant on the left side. Subtract 5 from both sides:\[-x + 5 - 5 = 10 - 5\]This reduces the equation to \(-x = 5\). Now, we need to solve for \(x\). To remove the negative sign from \(-x\), multiply both sides by \(-1\):\[-1(-x) = 5(-1)\]This results in \(x = -5\). By isolating the variable, we've accurately found the solution for \(x\).

Checking Solutions

After solving for the variable, it's crucial to check your solutions to ensure they satisfy the original equation. This step helps verify your calculations and confirms the accuracy. Substitute \(x = -5\) back into the original equation to see if it holds:\[(2(-5) + 1) + (4 - 3(-5)) = 10\]Perform the arithmetic within each grouping:- \(2(-5) + 1 = -10 + 1 = -9\) - \(4 - 3(-5) = 4 + 15 = 19\)So, combining the results gives:\[-9 + 19 = 10\]Both sides of the equation equal 10, confirming that \(x = -5\) is indeed the correct solution. Checking solutions is an essential step, always double-check your work.

Integer Solutions

In algebra, finding integer solutions means solving equations where the outcome for the variable is a whole number. Here, the term "integer" refers to numbers without fractional or decimal parts. In the exercise, the solution we discovered was \(x = -5\), which is an integer.
This often indicates simpler problem types, especially in educational contexts, as they allow students to comfortably focus on understanding fundamental algebraic techniques.

• Reality check: Ensure that your math operations maintain integer outcomes when the problem specifies or implies such.
• Useful tip: While solving equations, strive to keep the mental picture clear that unknowns should end up being clean, whole numbers if specified as integer solutions.

Remember, working towards integer solutions can sometimes hint at using specific algebraic strategies, like factoring whole numbers or finding common multiples.