Problem 31 - Prealgebra

Question

Solve each equation. Check your solution. $$y-7=-3$$

Step-by-Step Solution

Answer

The solution is \( y = 4 \).

Step 1: Isolate the Variable

To solve the equation \( y - 7 = -3 \), we must isolate \( y \). To do this, add 7 to both sides of the equation. This step gets rid of the -7 on the left side. \[y - 7 + 7 = -3 + 7\]

Step 2: Simplify the Equation

Simplify both sides of the equation after adding 7.The left side becomes just \( y \) because the -7 and +7 cancel each other out:\[y = -3 + 7\]The right side simplifies to 4:\[y = 4\]

Step 3: Check the Solution

To verify that \( y = 4 \) is correct, substitute 4 back into the original equation where \( y \) is.\[4 - 7 = -3\]Simplifying the left side:\[-3 = -3\]Since both sides of the equation are equal, our solution \( y = 4 \) is correct.

Key Concepts

Isolating the Variable

To solve any equation, the first essential step is to isolate the variable, which means getting the variable by itself on one side of the equation. In the equation given, \( y - 7 = -3 \), the variable is \( y \). You want to isolate \( y \) by removing any constants or coefficients that are affecting it. In this case, subtracting 7 makes it more complex for us. Therefore, to isolate \( y \), you perform the opposite operation—addition.

• Add 7 to both sides of the equation: \( y - 7 + 7 = -3 + 7 \)
• This results in \( y = 4 \).

By doing so, you've removed the constant influencing \( y \) and successfully isolated it. Now, \( y \) is all by itself, and we can see that \( y = 4 \). Isolating the variable is like understanding the core ingredient of a recipe; without it, you can't find the correct answer.

Simplifying the Equation

Once you've isolated the variable, simplifying the equation becomes straightforward. Simplification typically involves basic arithmetic or combining like terms. In our task at hand, simplifying was essential after we isolated the variable \( y \). What we did was:

• Calculating the right side of the equation: \(-3 + 7\).
• The operation yields \( 4 \), so our equation simplifies to \( y = 4 \).

Simplifying is just a fancy term for making things as straightforward as possible. It's about clearing out complexity to focus on what's needed. Think of it as tidying up your room to find what you're looking for. After isolation, simplification helps us see and claim our final answer, which in this particular problem is \( y = 4 \).

Checking the Solution

After finding a solution, it is absolutely crucial to verify its accuracy. This step ensures there are no mistakes in the calculations and boosts confidence that the solution is indeed correct. In our problem, we found that \( y = 4 \). To verify:

• Substitute \( 4 \) back into the original equation: \( y - 7 = -3 \)
• Replace \( y \) with \( 4 \): \( 4 - 7 \).
• Calculate: \( 4 - 7 = -3 \).

Since the original equation and the modified equation equal the same value, our solution holds true. This step, known as checking the solution, is akin to double-checking your work to ensure everything lines up properly—it confirms that you took the right path and ended up in the right spot. When both sides of the equation balance, you know you've achieved the correct result.