Problem 11 - Elementary Algebra

Question

Is the given value a solution to the linear equation? $$ 3 y-5=-2 y-15 ; y=-2 $$

Step-by-Step Solution

Answer

Yes, \( y = -2 \) is a solution.

Step 1: Substitute the given value into the equation

The initial linear equation provided is \( 3y - 5 = -2y - 15 \). We need to check if \( y = -2 \) is a solution. To do this, substitute \( y = -2 \) into the equation. This gives us \( 3(-2) - 5 = -2(-2) - 15 \).

Step 2: Simplify both sides of the equation

Start by simplifying both sides of the equation from Step 1.- Left side: \( 3(-2) - 5 = -6 - 5 = -11 \).- Right side: \( -2(-2) - 15 = 4 - 15 = -11 \).

Step 3: Compare the results of both sides

After simplifying both sides, we have:- Left side: \( -11 \)- Right side: \( -11 \)Since both sides are equal, \( y = -2 \) satisfies the equation.

Key Concepts

Solution Verification

Checking if a value is a solution to a linear equation is an important step in understanding algebra. To verify a solution, substitute the given value into the equation and simplify both sides. If both sides are equal, then the value is a solution; if not, then it isn't.
For example, in the equation:

• We substituted the value of y into the equation, replacing the variable with the given number. In our case, we substituted \( y = -2 \).
• After substitution, simplify the equation as much as possible.
• Finally, compare the results from both sides of the equation.

If both sides equal each other, the given value is confirmed as a valid solution, ensuring your answer is correct.

Algebraic Substitution

Algebraic substitution is a fundamental technique in solving equations. It involves replacing the variable in an equation with a given number, allowing for further calculations. This is useful in testing potential solutions.For our exercise:

• We started with the equation \( 3y - 5 = -2y - 15 \).
• Upon substituting \( y = -2 \), the equation becomes \( 3(-2) - 5 = -2(-2) - 15 \).
• This technique helps transition from a variable-based equation to a numerical one, making the verification process manageable.

This method is widely applicable across various types of problems and is essential for accurate solution testing.

Equation Simplification

Equation simplification is the process of condensing an equation to its simplest form. This step is crucial after substituting the variables to quickly determine if the equation holds true.In this exercise, we simplified:

• For the left side: \( 3(-2) - 5 \) became \( -6 - 5 \), which simplifies to \( -11 \).
• For the right side: \( -2(-2) - 15 \) became \( 4 - 15 \), which also simplifies to \( -11 \).

After simplifying both sides, ensure they equate to confirm the proposed solution. This process is key in verifying consistency and correctness in algebraic solutions.