Problem 9 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$12 x+3=-2 x+17$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1\).

Step 1: Combine like terms

We start with the equation: \(12x + 3 = -2x + 17\). To solve for \(x\), first combine the like terms by bringing all the \(x\) terms to one side. Add \(2x\) to both sides to get all \(x\) terms on the left side: \(12x + 2x + 3 = 17\). This simplifies to \(14x + 3 = 17\).

Step 2: Isolate the x-term

Next, isolate the \(x\)-term by removing the constant on the left side. Subtract 3 from both sides of the equation: \(14x + 3 - 3 = 17 - 3\). The equation simplifies to \(14x = 14\).

Step 3: Solve for x

Now that we have \(14x = 14\), solve for \(x\) by dividing both sides of the equation by 14: \(x = \frac{14}{14}\). This gives us \(x = 1\).

Key Concepts

Combining Like Terms

When solving linear equations, one of the first skills you'll need is the ability to combine like terms. These terms are similar because they contain the same variables raised to the same power. In the equation given, we start with combining like terms to simplify the problem. The expression on either side of the equation must be consolidated to make the solving process straightforward. For instance, you have terms involving \(x\) on both sides: \(12x\) and \(-2x\). By moving \(-2x\) to the left side through addition, we get \(12x + 2x\). Combining these gives us \(14x\).The goal is to have terms involving the variable you're solving for, \(x\) in this case, lined up on one side of the equation. As you practice, remember:

• Combine constants separately from variables.
• Ensure each side of the equation is simplified before proceeding to the next step.

Isolating Variables

Once like terms have been combined, the next step is isolating the variable. It's a key part of solving equations as it leads to discovering the variable's value.In the example equation, after combining like terms, we have \(14x + 3 = 17\). The next task is to get \(x\) by itself on one side of the equation. This involves removing any constants or coefficients. Here, it means subtracting 3 from both sides. This gives \(14x = 14\).To successfully isolate variables:

• Use subtraction or addition to eliminate constants associated with your variable.
• Always perform the same operation on both sides of the equation to maintain equality.

By isolating the variable, you transform a complex equation into something much simpler.

Step-by-Step Solution

Understanding how to break down problems into simpler steps is crucial in solving equations effectively. The step-by-step approach ensures no detail is overlooked, making complex problems manageable.For our equation, the first step was combining like terms, which transformed \(12x + 3 = -2x + 17\) into \(14x + 3 = 17\). By carefully managing each side of the equation, we simplified it effectively.After which, we isolated the \(x\) variable by removing the constant \(3\), resulting in \(14x = 14\). Finally, we solved for \(x\) by dividing both sides by 14, getting \(x = 1\).Following a structured approach involves:

• Clarifying each operation before performing it.
• Double-checking each step to ensure calculations are accurate.
• Being methodical helps in avoiding mistakes and builds confidence in problem-solving.

This step-by-step process not only simplifies the task but also reinforces your understanding of how equations work.