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Chapter 4: Problem 25

Solve each system of equations. $$\begin{aligned}x+y &=7 \\\y-z &=-1 \\\x &+3 z=18\end{aligned}$$

Short Answer

Expert verified
The solution is \( x = 3 \), \( y = 4 \), and \( z = 5 \).

Step by step solution

01

Express one variable in terms of another from the first equation

From the first equation, we have: \[ x + y = 7 \]Express y in terms of x: \[ y = 7 - x \]
02

Substitute y in the second equation

Substitute \( y = 7 - x \) into the second equation \[ y - z = -1 \]We get: \[ (7 - x) - z = -1 \]Rearranging gives: \[ 7 - x - z = -1 \]Simplify to find z: \[ z = 8 - x \]
03

Substitute x and z into the third equation

Now substitute \( z = 8 - x \) into the third equation \[ x + 3z = 18 \]We get: \[ x + 3(8 - x) = 18 \]Solve for x: \[ x + 24 - 3x = 18 \]\[ -2x + 24 = 18 \]\[ -2x = -6 \]\[ x = 3 \]
04

Substitute x back to find y and z

Using \( x = 3 \):From \( y = 7 - x \)\[ y = 7 - 3 \]\[ y = 4 \]And from \( z = 8 - x \)\[ z = 8 - 3 \]\[ z = 5 \]

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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 a powerful tool for solving systems of equations where we replace one variable with an expression containing another variable. This technique simplifies the system step-by-step, making it easier to solve.
Here鈥檚 how it works:
  • First, solve one of the equations for one variable in terms of the others. In our example, from the equation \(x + y = 7\), we solved for \(y\) to get \(y = 7 - x\).
  • Next, substitute this expression into the other equations. By replacing \(y\) with \(7 - x\) in the equation \(y - z = -1\), it simplifies to \(7 - x - z = -1\).
This substitution method eliminates \(y\) and lets us focus on solving for the remaining variables. As shown in the solution process, this method effectively reduces the complexity, making solving the equation an easier task.
Linear Equations
Linear equations are fundamental in algebra and can be recognized by their standard form, which is \(ax + by + cz = d\), where \(a, b, c\), and \(d\) are constants.
Let鈥檚 understand their properties:
  • Each term is either a constant or the product of a constant and a single variable.
  • These equations graph as straight lines when plotted on a graph.
In the example, we have three linear equations:
  • \(x + y = 7\)
  • \(y - z = -1\)
  • \(x + 3z = 18\)
All these fit the linear equation framework. By manipulating and simplifying them, we solve for the variables \(x\), \(y\), and \(z\). Understanding linear equations is essential to grasp the concept of solving systems of equations.
Algebra Problem Solving
Algebra problem solving often involves a multi-step process where strategic thinking and systematic approaches are necessary. To master this, consider the following steps:
  • Identify: Recognize what you need to find. Here, we need to find the values of \(x\), \(y\), and \(z\).
  • Simplify: Break down complex problems into simpler steps. For instance, we first expressed \(y\) in terms of \(x\) and then substituted back to find \(z\).
  • Substitute: Use solved parts to reduce the system to simpler forms, just like we substituted \(y = 7 - x\) and \(z = 8 - x\) into the remaining equation.
  • Solve: Isolate the variable to find its value. Solving \(x\) in \(x + 3(8 - x) = 18\) gave us \(x = 3\).
  • Verify: Always plug your solutions back into the original equations to ensure they work. Such as verifying \(x = 3, y = 4, z = 5\) solve all initial equations correctly.
Problem-solving in algebra becomes manageable when you follow these structured steps, ensuring every part of the equation is addressed accurately.

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