Question

Solve the system of linear equations using the substitution method. \(4 x+y+5 z=5\) \(8 x+2 y+10 z=10\) \(x-y-2 z=-2\)

Short answer

The solution of the system is \(x = y + 2z -2 \), \(y= 3/5 - (13/5)z\), and z as any real number. Since the system has infinite solutions, by choosing any value for \(z\), \(x\) and \(y\) can be calculated. This system of equations is dependent.

Step-by-step solution

Checking for Proportional Equations

Check if any of the equations are proportional to each other. It appears that the first and second equation are proportional. If we divide the coefficients in the second equation by 2, we get exactly the coefficients in the first equation. This means that the first two equations are representing the same line.

Solving One Equation for One Variable

Select the third equation to solve for one variable. Solving the third equation \(x-y-2 z=-2\) for \(x\) gives \(x=y+2 z-2\)

Substituting the Expression into Other Equations

Substitute \(x=y+2 z-2\) into the first equation \(4 x+y+5 z=5\), we get \(4(y+2 z-2)+y+5 z=5\). This simplifies to \(5y+13z-3=0\)

Solving for Remaining Variables

Now, simplify the equation got from substitution and solve for y and z. This results in \(y= 3/5 - (13/5)z\). Now substitute y back to the equation x = y + 2z - 2 and solve for z, this should provide the answer.

Key concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. In our exercise, the system can be represented as the following equations:

• \( 4x + y + 5z = 5 \)
• \( 8x + 2y + 10z = 10 \)
• \( x - y - 2z = -2 \)

Each equation represents a plane in three-dimensional space. The point where all planes intersect is the solution to the system. If two equations are proportional, they represent the same plane, which simplifies our calculations as seen in our exercise. Identifying proportional equations first can help reduce unnecessary steps later on.

Solving Equations

The process of solving equations involves finding values for the variables that make all the equations true. In this exercise, we use the substitution method. By substituting one variable's expression into another equation, we reduce the number of variables, simplifying the equations.The key steps to solve equations through substitution in this problem are:

• Check if any equations are proportional to eliminate redundancy.
• Solve one equation for a single variable. For example, \( x = y + 2z - 2 \).
• Substitute this expression into another equation to find relationships between remaining variables.

This strategy streamlines the problem-solving process, allowing us to find the solutions step by step.

Algebra 2

Algebra 2 builds on the concepts learned in Algebra 1, introducing more complex equations and systems. It often involves systems like the one in this exercise, where you work with multiple equations simultaneously. Key concepts to remember include:

• Understanding how to manipulate and solve for different variables.
• Recognizing patterns, such as proportional relationships between equations.
• Using methods like substitution to simplify systems and solve them efficiently.

In Algebra 2, these techniques are fundamental for solving real-world problems and advancing in mathematics.

Proportional Equations

Proportional equations have a significant role in simplifying systems of equations. When two equations within a system are proportional, it means they essentially describe the same line or plane.In our example:

• The first equation \( 4x + y + 5z = 5 \) and the second equation \( 8x + 2y + 10z = 10 \) are proportional.
• This is evident by comparing coefficients: dividing each term in the second equation by 2 yields the first equation.

This discovery allows us to remove redundant equations from the system, minimizing complexity. Recognizing proportionality is a critical skill for solving larger and more complex systems efficiently.