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Chapter 3: Problem 33

Solve each system. If a system's equations are dependent or if there is no solution, state this. $$ \begin{aligned} &x+y+z=83\\\ &y=2 x+3\\\ &z=40+x \end{aligned} $$

Short Answer

Expert verified
The solution is \( x = 10 \), \( y = 23 \), and \( z = 50 \).

Step by step solution

01

Substitute y from the second equation

The second equation gives \( y = 2x + 3 \). We can substitute this expression for y in the first equation.
02

Substitute z from the third equation

The third equation gives \( z = 40 + x \). We can substitute this expression for z in the first equation.
03

Combine and simplify the first equation

Substitute \( y = 2x + 3 \) and \( z = 40 + x \) into the first equation: \( x + (2x + 3) + (40 + x) = 83 \). Simplify the equation: \( 4x + 43 = 83 \).
04

Solve for x

Subtract 43 from both sides of the equation: \( 4x = 40 \). Then, divide by 4: \( x = 10 \).
05

Solve for y

Substitute \( x = 10 \) back into the equation for y: \( y = 2(10) + 3 = 23 \).
06

Solve for z

Substitute \( x = 10 \) back into the equation for z: \( z = 40 + 10 = 50 \).
07

Verify the solution

Substitute \( x = 10 \), \( y = 23 \), and \( z = 50 \) back into the original equations to ensure they hold true. \( 10 + 23 + 50 = 83 \), \( 23 = 2(10) + 3 \), and \( 50 = 40 + 10 \) are all correct.

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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 used to solve systems of linear equations by expressing one variable in terms of another. This method is crucial when equations are interdependent and direct elimination isn't possible.
For example, from the given system, we have:
  • Second equation: \( y = 2x + 3 \)
  • Third equation: \( z = 40 + x \)
These equations allow you to express both \( y \) and \( z \) in terms of \( x \).
Once you substitute these equations into the first one, the system is simplified, making it easier to solve for \( x \), and consequently, for \( y \) and \( z \). The substitution method helps break down complex problems into manageable steps.
dependent systems
Dependent systems occur when the equations in a system describe the same line or plane, meaning they are not unique but multiples of one another. In other words, solutions to these systems form a line or a plane of solutions.
In our exercise, if we had ended up with equivalent equations after substituting, we might have recognized a dependent system. For instance, equations aligning to forms like \(x + y = 1\) and \(2x + 2y = 2\) illustrate dependency since they represent the same line.
It's important to identify dependent systems as they indicate an infinite number of solutions, contrasting systems with unique solutions or no solution, like parallel lines with no intersection.
linear equations
Linear equations are polynomial equations of the first degree, which graph as straight lines. These equations typically take the form \( ax + by + cz = d \), where \( a \), \( b \), and \( c \) are coefficients and \( x \), \( y \), and \( z \) are variables. The solutions to these equations can be visualized geometrically as points of intersection on the coordinate plane or space.
In the provided exercise, each equation is linear:
  • First equation: \( x + y + z = 83 \)
  • Second equation: \( y = 2x + 3 \)
  • Third equation: \( z = 40 + x \)
These forms allow us to use methods like substitution or elimination for solutions. Understanding linear equations is foundational to grasping more complex systems and mathematical concepts.

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