Problem 16 - Algebra and Trigonometry

Question

Solve and check each linear equation. $$\begin{aligned}&45-[4-2 y-4(y+7)]=\\\&-4(1+3 y)-[4-3(y+2)-2(2 y-5)]\end{aligned}$$

Step-by-Step Solution

Answer

The solution to the equation is \( y = -61/19 \).

Step 1: Simplify the Equation

Start by simplifying each side of the equation. Distribute any multipliers and combine like-terms. The equation now reads: \[ 45 - 4 + 2y + 4y + 28 = -4 - 12y - 4 + 3y + 6 - 4y + 10 \].

Step 2: Reorder and Group Like Terms

Next, group like terms and re-order the equation for easier manipulation. This would turn the equation into: \[ 45 - 4 + 28 + 2y + 4y = -4 - 4 + 6 - 12y + 3y - 4y +10 \] .

Step 3: Simplify Further

Simplify each side of the equation further, adding and subtracting integer terms together, and combining like terms: \[ 69 + 6y = -2 - 13y +10 \]. Simplify further into: \[ 69 + 6y = 8 - 13y \].

Step 4: Remove Constants on One Side

Next, remove the constants from the left side of the equation to isolate the variable term by subtracting 69 from each side. This will present: \[ 6y = -61 - 13y \].

Step 5: Isolate the variable

Our final step is to isolate the variable term y on one side of the equation by moving -13y over to the left hand side resulting to: \[ 6y + 13y = -61 \]. Which further simplifies to: \[ 19y = -61 \]. Divide each side by 19 to solve for y: \[ y = -61/19 \].

Step 6: Check the Solution

To verify, we replace the y in original equation with -61/19. If the left-hand side equals the right-hand side after simplifying, then the solution is correct.

Key Concepts

Solving Linear Equations

Linear equations involve finding the value of the variable that makes the equation true. They typically look like \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable.

• To solve a linear equation, we aim to "isolate" the variable on one side of the equation. This often involves performing the same operation on both sides.
• In our problem, the goal is to find the value of \( y \) that balances both sides of the equation. We apply operations like addition, subtraction, or factoring to both sides to achieve this.

Here's a useful trick: order of operations is crucial. Start simplifying where it's most complicated, often by first handling parentheses and exponents. This organizes our equation and makes it easier to further manipulate during our solving process.
Linear equations are fundamental in many areas of mathematics because they describe a relation where each value of \( x \) gives exactly one corresponding value of \( y \). Understanding the solving process strengthens your grasp of mathematical logic.

Simplifying Expressions

Simplifying expressions is a crucial step in solving any algebraic equation, and it involves making an equation shorter and easier to handle without changing its value. To simplify an expression, focus on:

• Distribution: Applying multiplication over addition or subtraction inside parentheses. For example, distributing \(4\) in \(4(y+7)\) gives \(4y + 28\).
• Combining Like Terms: Grouping and simplifying similar terms such as \(2y\) and \(4y\).

Simplification in the given problem started with rearranging and combining like terms on both sides of the equation, transforming them into a more concise form. Once terms are simplified, it's easier to solve for the variable since the equation is less cluttered, making the path to the solution more apparent.
Remember: Simplification doesn't change the solution but rather clarifies your path to achieving it.

Checking Solutions

Checking solutions in linear equations is a vital step to ensure the accuracy of the solution obtained. It's basically re-substituting the value of the variable back into the original equation to confirm that it satisfies the equation.

• After determining \( y = -61/19 \), we substitute \( y \) back into the original equation.
• Simplifying both the left-hand and right-hand sides should give you the same value.

This check not only confirms your solution is correct but also enhances understanding of the relationship between the expressions and their outcomes.
While checking might seem tedious, it's a practice that builds confidence and reinforces the concepts behind solving linear equations. By verifying each step of the solution, misunderstandings or errors in manipulation can be easily identified and corrected.