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2.2. Solving Equations having the Variable on both Sides
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Today, we're going to learn about solving equations with variables on both sides. Can anyone give an example of such an equation?
How about 2x - 3 = x + 2?
Exactly! In this case, both sides contain the variable x. The goal is to isolate the variable. Can anyone tell me how we might begin solving this?
We can subtract x from both sides?
Good thinking! That's a great first step. When we subtract x, what do we get on each side?
We get 2x - x - 3 = 2.
Right! Now, can someone simplify this further?
That gives us x - 3 = 2.
Excellent! Now, what’s the next step?
Add 3 to both sides, so we get x = 5.
Perfect! Remember, we always need to do the same operation on both sides to keep the equation balanced.
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Let’s look at a more complex example: 5x + 7/2 = x - 14. What’s our first step?
We could multiply by 2 to eliminate the fraction!
Exactly! Multiplying both sides by 2 gives us a simpler equation. What do we get?
That becomes 10x + 7 = 2x - 28.
Great! Now, how do we isolate x from here?
We can subtract 2x from both sides.
Correct! And what does that lead us to?
It simplifies to 8x + 7 = -28.
Now, what’s our next move?
We subtract 7 from both sides to get 8x = -35.
Fantastic! Finally, how do we find x?
We divide by 8, giving us x = -35/8.
Absolutely right! This method of manipulating both sides is key in solving these equations.
Overview
Short Summary
This section focuses on solving equations with variables on both sides, detailing methods and providing examples.
Medium Summary
In this section, we learn how to solve equations where variables are present on both sides. We explore various methods to manipulate these equations, illustrated by examples that show the step-by-step process to isolate the variable and obtain solutions.
Detailed Summary
Solving Equations with the Variable on Both Sides
An equation represents the equality of the values of two expressions, such as in the example 2x - 3 = 7. Here, the left-hand side (LHS) and the right-hand side (RHS) can also include variables, as demonstrated in 2x - 3 = x + 2.
To solve these equations, we need to isolate the variable. The section introduces how to manipulate both sides of the equation. For example, in the equation 2x - 3 = x + 2, steps such as subtracting x from both sides help us to eventually isolate x as shown in the solution where it simplifies to x = 5.
Additional examples, such as 5x + 7/2 = x - 14, illustrate multiplying each side of the equation to simplify the expressions further. The section emphasizes understanding the operations performed on both sides to maintain equality and reach the solution effectively.
Example 2: Solve
Solution: We have
or
or
(solution)
Reference YouTube Videos
Audio Book
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Create a free accountAn equation is the equality of the values of two expressions. In the equation 2x – 3 = 7, the two expressions are 2x – 3 and 7. In most examples that we have come across so far, the RHS is just a number. But this need not always be so; both sides could have expressions with variables. For example, the equation 2x – 3 = x + 2 has expressions with a variable on both sides; the expression on the LHS is (2x – 3) and the expression on the RHS is (x + 2).
Detailed Explanation
In mathematics, an equation states that two expressions are equal to each other. For instance, in the equation given (2x - 3 = x + 2), we have a variable (x) on both sides of the equation. This means that we need to find a value for x that makes both the left-hand side (LHS) and right-hand side (RHS) equal. Unlike simpler equations where one side might just be a number, these types of equations require us to manipulate the expressions to isolate the variable.
Examples & Analogies
Imagine you have two jars of candies. The first jar has a variable number of candies minus a few, represented by the expression 2x - 3, while the second jar has a different expression representing a variable number of candies plus a couple (x + 2). To find out how many candies are in each jar, you must figure out the value of x that makes both jars have the same amount.
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Create a free accountExample 1: Solve 2x – 3 = x + 2. Solution: We have 2x = x + 2 + 3 or 2x = x + 5 or 2x – x = 5 (subtracting x from both sides) or x = 5 (solution). Here we subtracted from both sides of the equation, not a number (constant), but a term involving the variable.
Detailed Explanation
To solve the equation 2x - 3 = x + 2, we want to get all terms involving x on one side and constants on the other. Starting with the equation, we move -3 over to the right side, which gives us 2x = x + 5. To isolate x, we subtract x from both sides, giving us x = 5. This means that if we replace x with 5 in the original equation, both sides will equal each other, confirming it's the correct solution.
Examples & Analogies
Think of this as an evenly balanced scale. You start off with some weights on one side (2x - 3) and want to balance them with weights on the other side (x + 2). By doing proper adjustments (subtracting), you eventually find the exact weight (value of x) that balances the scale.
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Create a free accountExample 2: Solve 5x + = x −14. Solution: Multiply both sides of the equation by 2. We get (2×(5x + 7/2)) = (2×(x − 14)). After simplification, we reach 10x + 7 = 3x - 28.
Detailed Explanation
The equation 5x + 7 = x - 14 includes fractions. To eliminate the fraction, we can multiply the entire equation by 2 (the denominator). This gives us integers throughout, making it easier to solve. After multiplying and simplifying, we isolate x by moving 3x to the left side by subtracting it, yielding 7x + 7 = -28. Finally, we solve for x to get x = -5.
Examples & Analogies
Imagine a baking recipe where you need to double all ingredients. The term with x in your recipe represents the number of cups of flour. By multiplying the entire recipe by 2 (just like we multiplied the equation), you ensure that everything stays proportional. By the end, you find exactly how many cups you need, represented by the solution for x.
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Isolating Variables: The process of getting the variable on one side of the equation.
Maintaining Equality: Understanding that operations must be balanced on both sides.
Step-by-Step Solutions: Breaking down each operation to find the solution.
Examples
Memory Aids
Interactive tools to help you remember key concepts