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Chapter 7: Problem 39

Solve each equation. $$-350+z=125$$

Short Answer

Expert verified
The solution is \(z = 475\).

Step by step solution

01

Arrange the Equation

Start with the given equation:\[-350 + z = 125\] The goal is to solve for \(z\).
02

Isolate the Variable

Add 350 to both sides of the equation to isolate \(z\): \[-350 + z + 350 = 125 + 350\] This simplifies to: \[z = 475\]
03

Verification

Substitute \(z = 475\) back into the original equation to check: \[-350 + 475 = 125\] This is a true statement, confirming that \(z = 475\) is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Variable
When solving linear equations, one of the first and most crucial steps is isolating the variable. This means we want to get the variable (in this case, \(z\)) alone on one side of the equation.
To do this, you need to "move" everything else to the other side. This involves using basic algebraic operations like addition, subtraction, multiplication, or division.
In our example of the equation \(-350 + z = 125\):
  • We have a \(-350\) added to \(z\). To isolate \(z\), we need to counteract that \(-350\) by adding 350 to both sides of the equation.
  • This step changes the equation to \(z = 125 + 350\), which simplifies to \(z = 475\).
Isolating the variable is fundamental because it helps us find the value of the variable that makes the equation true. Every other operation in solving equations centers on this step. Keep practicing it until it becomes second nature!
Addition Property of Equality
The addition property of equality is an algebraic principle that lets us add the same number to both sides of an equation without changing the equality.
Imagine you're balancing a scale. If you add the same weight to both sides, it stays balanced. This is exactly what we're doing with equations.
In the specific equation \(-350 + z = 125\):
  • We use the addition property of equality by adding 350 to both sides, which helps us isolate \(z\).
  • This results in \(z = 125 + 350\).
This property is powerful because it keeps our equation balanced while we manipulate it to solve for the variable. It's like ensuring both sides of the equation are always treated equally, which is key to accurate problem-solving in algebra.
Verification of Solutions
After solving an equation, it's always a smart idea to verify your solution. Verification helps confirm that the solution is correct and that no errors were made during any of the steps.
In our example, once we found \(z = 475\), we wanted to ensure our calculation was accurate by substituting \(475\) back into the original equation:
  • We replaced \(z\) with \(475\) and checked if \(-350 + 475\) equals 125.
  • Since it did, the equation \,\(-350 + 475 = 125\,\) holds true.
Verification is an important last step in solving equations. It is the best way to be sure of the correctness of your solution. Keep this habit to build confidence in your computations!

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