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Chapter 8: Problem 50

Solve for \(z\) if \(x=1\) and \(y=-2\) $$ -3 x-y+z=1 $$

Short Answer

Expert verified
z = 2

Step by step solution

01

Substitute the given values

Substitute the given values of \(x = 1\) and \(y = -2\) into the equation -3x - y + z = 1.
02

Simplify the equation

After substitution, the equation becomes -3(1) - (-2) + z = 1 Simplify the terms: -3 + 2 + z = 1.
03

Solve for \(z\)

Combine the constants: -1 + z = 1 Add 1 to both sides to isolate \(z\): z = 2.

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Key Concepts

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

substitution method
The substitution method is vital in solving systems of linear equations and individual equations with given values. This method allows us to replace one variable with its actual value, making the equation simpler to solve. In our exercise, the values of two variables, \(x=1\) and \(y=-2\), are given.
We substitute these values into the equation \[-3x - y + z = 1\]. This helps in converting the original equation with multiple variables into an equation with just one variable. This makes it easier to handle and solve.
This method is not only useful for solving linear equations but is also a crucial technique in solving systems of equations where we need to find multiple unknowns.
algebraic simplification
Once we have substituted the values, the next step is to simplify the equation. This involves combining like terms and performing arithmetic operations to make the equation more straightforward. Here is how we do it:
After substituting \(x = 1\) and \(y = -2\) into the equation \[-3(1) - (-2) + z = 1\], we perform the multiplications and adjust the signs:
  • -3 \(\times\) 1 = -3
  • Double negative turns positive: -(-2) becomes 2
Now, our equation looks like \[-3 + 2 + z = 1\]. This step reduces the complexity of our problem, and now we are left with easier terms to deal with. Finally, we combine the constants which simplifies our equation to \[-1 + z = 1\].
isolating variables
Isolating a variable means getting the variable by itself on one side of the equation. It is a critical step in solving equations. To isolate \(z\) in the equation \[-1 + z = 1\], we need to remove the constant term on its side.
We can do this by adding 1 to both sides of the equation. Adding the same number to both sides keeps the equality balanced:
\[-1 + z + 1 = 1 + 1\] Simplifying both sides, we get: \[z = 2\]
Now, \(z\) is isolated and we have our solution! This step ensures that all other terms are cancelled out, leaving us with the variable we set out to solve for.

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