Problem 44 - Algebra 1

Question

Solve the equation. $$7 z+30=-5$$

Step-by-Step Solution

Answer

The solution to the equation \(7 z + 30 = -5\) is \( z = -5\).

Step 1: Simplify the Equation

We can start by moving the 30 from the left side of the equation to the right side. This can be done by subtracting 30 from both sides of the equation. As a result, the equation now becomes: \(7z = -5 - 30.\)

Step 2: Perform the Subtraction on the Right

Now, perform the subtraction operation of -5 - 30 which results in -35. So, our equation now becomes: \(7z = -35.\)

Step 3: Solve for 'z'

Finally, solve for 'z' by dividing both sides of the equation by 7. This results in \(z = -35/7\).

Step 4: Simplify the Results

When we simplify \( -35/7 \), we get \( z = -5\).

Key Concepts

Equation Simplification

The process of equation simplification is an essential step in solving mathematical problems. It involves reducing an equation to its simplest form, making it easier to work with and understand. When we encounter a linear equation like \(7z + 30 = -5\text{,}\) our primary goal is to isolate the variable \(z\) on one side of the equation. Simplifying the equation starts with eliminating any additions or subtractions around the variable. In this case, we see the number 30 added to \(7z\), which doesn't help us see what one \(z\) equals.

To simplify, we perform the inverse operation to both sides. Since we have a positive 30 added to \(7z\), we subtract 30 from both sides. This doesn't change the equality because we're doing the same thing to each side, it's like balancing scales. By doing so, we move closer to having \(z\) stand alone, and the equation becomes much less cluttered and more manageable. Remember, keep the balance, and you'll maintain the truth of the equation as you simplify it.

Subtraction in Equations

Subtraction in equations is like telling a story backwards. If you know the ending, you want to find out how the story began. Imagine our equation is the final chapter where \(7z\) has somehow been combined with 30 to result in -5. Our job is to retrace the steps. By subtracting 30 from each side of the equation \(7z + 30 = -5\) during the simplification process, we're taking away the same 'obstacle' from both sides, ensuring the 'balance' of the equation is maintained.

After subtracting, we get \(7z = -5 - 30\), which further simplifies to \(7z = -35\). This subtraction step is crucial because it paves the way to getting \(z\) isolated. It peels away the layers of the equation until the core, the variable we're solving for, is exposed. Just like peeling an onion, after you remove enough layers (in this case, subtracting 30), you get closer to the core (the variable \(z\)).

Isolating Variables

Isolating variables is like finding the key that unlocks the mystery of the equation. It's the final step where we get to discover the value of the unknown character in our mathematical story. Once we have \(7z = -35\) from our previous steps, we notice that \(z\) is not alone yet. It's still being multiplied by 7. To isolate \(z\), we want to do the reverse of multiplying; we divide.

By dividing both sides of the equation by 7, we counteract the multiplication, effectively detaching \(z\) from any other numbers. After this division, we're left with \(z = -35/7\), which simplifies down to \(z = -5\). We now have \(z\) completely isolated, meaning it's all by itself, and we've found the key that unlocks the entire equation. Most importantly, the equation remains balanced because just like any good story, everything has to make sense from beginning to end.