Solving Equations with Variables on Both Sides
When faced with equations that feature variables on both sides, the challenge lies in isolating the variable to find its value. These types of equations can initially seem intimidating; however, with the right strategies and techniques, they can be tackled effectively. In this article, we will explore methods for manipulating these equations, simplifying them, and ultimately solving for the variable.
Understanding Equations with Variables on Both Sides
Let's get right into it—an equation with variables on both sides is one that contains a variable on the left and another variable on the right. For example:
\[ 3x + 5 = 2x + 10 \]
Your goal is to rearrange this equation so that all terms containing the variable (in this case, \(x\)) are on one side and the constants are on the other side. The primary operations we can utilize to accomplish this are addition, subtraction, multiplication, and division.
Step-by-Step Process for Solving
Let’s break down the process step by step.
Step 1: Identify the Variables and Constants
In our example equation \(3x + 5 = 2x + 10\):
- Variables: \(3x\) (from the left side) and \(2x\) (from the right side)
- Constants: \(5\) (left) and \(10\) (right)
Step 2: Move Variable Terms to One Side
The first step is to get all variable terms on one side of the equation. You can do this by performing the same operation on both sides of the equation. In our case, let’s move \(2x\) from the right side to the left side by subtracting \(2x\) from both sides:
\[ 3x + 5 - 2x = 2x + 10 - 2x \]
This simplifies to:
\[ x + 5 = 10 \]
Step 3: Move Constant Terms to the Other Side
Next, we need to isolate \(x\) by moving the constants. To do this, subtract \(5\) from both sides:
\[ x + 5 - 5 = 10 - 5 \]
This gives us:
\[ x = 5 \]
Step 4: Check Your Solution
It’s always a good idea to check your solution by substituting the value back into the original equation. Let's verify:
Substituting \(x = 5\) into the original equation:
\[ 3(5) + 5 = 2(5) + 10 \]
Calculating each side:
\[ 15 + 5 = 10 + 10 \]
Both sides equal \(20\), so our solution is confirmed!
Example Problems to Practice
Let’s work through a couple more examples to solidify your understanding.
Example 1:
Solve the equation:
\[ 4y - 3 = y + 7 \]
Step 1: Move \(y\) to the left side.
\[ 4y - 3 - y = y + 7 - y \]
This simplifies to:
\[ 3y - 3 = 7 \]
Step 2: Move the constant to the right side.
\[ 3y - 3 + 3 = 7 + 3 \]
This gives:
\[ 3y = 10 \]
Step 3: Solve for \(y\).
\[ y = \frac{10}{3} \]
Step 4: Check your solution.
Substituting back:
\[ 4\left(\frac{10}{3}\right) - 3 \stackrel{?}{=} \frac{10}{3} + 7 \]
Calculating both sides confirms that the equation balances, so \(y = \frac{10}{3}\) is correct.
Example 2:
Now, try this equation:
\[ 5a + 8 = 3(a - 2) + 4 \]
Step 1: Distribute on the right side.
This becomes:
\[ 5a + 8 = 3a - 6 + 4 \]
Which simplifies to:
\[ 5a + 8 = 3a - 2 \]
Step 2: Move \(3a\) to the left.
\[ 5a - 3a + 8 = -2 \]
Simplifying gives:
\[ 2a + 8 = -2 \]
Step 3: Move the constant to the right.
\[ 2a + 8 - 8 = -2 - 8 \]
This simplifies to:
\[ 2a = -10 \]
Step 4: Solve for \(a\).
\[ a = -5 \]
Step 5: Verify the solution.
Substituting \(a = -5\) back into the original equation:
\[ 5(-5) + 8 \stackrel{?}{=} 3(-5 - 2) + 4 \]
Calculating both sides verifies the balance and confirms that \(a = -5\) is indeed the correct solution.
Working with Like Terms
While isolating variables, it’s also crucial to be adept at combining like terms. Like terms are terms that have the same variable factors with the same exponents. For example, \(2x\) and \(3x\) are like terms, while \(2x\) and \(3y\) are not.
Combining Like Terms Example
If you have an equation like:
\[ 3x + x - 2x + 5 = 2x + 10 \]
Step through as follows:
-
Combine \(3x + x - 2x\) on the left: \[ (3 + 1 - 2)x + 5 = 2x + 10 \] This simplifies to: \[ 2x + 5 = 2x + 10 \]
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From here, you can see that \(2x\) cancels on both sides, resulting in: \[ 5 = 10 \] Since this is not true, it means there are no solutions to this equation—it is inconsistent.
Conclusion
Solving equations with variables on both sides doesn't have to be daunting. By moving terms strategically and combining like terms, you can isolate your variable and arrive at the solution with clarity. Remember to check your work for accuracy, and with practice, you'll become more adept at these types of algebraic manipulations. Keep practicing, and soon you'll find that solving these equations becomes second nature!