Problem 54 - Algebra A Combined Function

Question

\(10 y+15=-5\)

Step-by-Step Solution

Answer

The solution is \(y = -2\).

Step 1: Understanding the Equation

The equation given is \(10y + 15 = -5\). Our goal is to solve for the variable \(y\). We need to isolate \(y\) on one side of the equation.

Step 2: Subtract 15 from Both Sides

To isolate the term with the variable \(y\), we first get rid of the constant term on the left side of the equation. Subtract 15 from both sides: \(10y + 15 - 15 = -5 - 15\). This simplifies to \(10y = -20\).

Step 3: Divide by 10

Now, divide both sides of the equation by 10 to solve for \(y\). So, \(\frac{10y}{10} = \frac{-20}{10}\). This results in \(y = -2\).

Step 4: Verify the Solution

To ensure that our solution is correct, substitute \(y = -2\) back into the original equation: \(10(-2) + 15 = -5\). Simplifying the left side gives \(-20 + 15 = -5\), which is true. Thus, \(y = -2\) is the correct solution.

Key Concepts

Solving Equations Step-by-Step

To tackle linear equations like the one given, it's essential to approach them systematically. Linear equations have variables, like our unknown "y," which we need to solve. A step-by-step process means breaking down each part of the equation until we've isolated our variable.

• **Understand the problem:** Begin by comprehensively understanding the given equation. For our example, the equation is \(10y + 15 = -5\).


• **Reorder and Simplify:** Start by reorganizing the equation to focus on getting the variable by itself on one side. This plan requires removing constants or numbers that don't include the variable.


• **Perform operations carefully:** Follow arithmetic operations precisely. Always subtract or add terms on both sides to maintain balance. For instance, after subtracting 15 from both sides of \(10y + 15 = -5\), we get \(10y = -20\).

Approaching problems step-by-step also minimizes errors. By doing this, you can confidently work your way toward a solution.

Isolate the Variable

The crux of solving equations is isolating the variable, which in our case is "y." What does this mean? Simply put, it means getting "y" by itself on one side of the equation, often with a coefficient of 1.To isolate "y":

• **Remove known numbers:** Begin by eliminating any constants. From our example, we moved the "15" by subtracting it, leading us to \(10y = -20\).


• **Deal with coefficients:** Then, make sure "y" stands alone. Here, we do this by dividing each side of \(10y = -20\) by 10, giving us \(y = -2\).

This method ensures clarity and solves the equation successfully. Isolating the variable is the heart of equation-solving, helping you directly find the unknown value.

Verify the Solution

Never forget to check your work by verifying your solution. Verifying your solution is crucial to ensure that mistakes have not been made along the way.Here's how you verify:

• **Substitute back:** Plug the obtained value of the variable back into the original equation. For our case, substituting \(y = -2\) into \(10y + 15 = -5\) enables us to double-check our work.


• **Calculate and confirm:** Simplify each side. When we substitute \(y = -2\), we calculate \(10(-2) + 15\) and find it equals \(-5\). This confirms the left side matches the original right side of the equation.


• **Confidence in correctness:** Conclude by confirming the solution works. If both sides equal, like our equation does, the solution, here \(y = -2\), is verified as accurate.

Verification isn’t just a formality—it's a vital step in ensuring your solution stands up to scrutiny.