Question

What is the solutions of the following simultaneous equations? \(x+y+z=6, \quad x+2 y+3 z=14, \quad x+3 y+z=10\) (a) \(x=1, y=2, z=4\) (b) \(x=1, y=2, z=3\) (c) \(x=2, y=1, z=3\) (d) none of these

Short answer

Question: Determine the values of x, y, and z that satisfy the given system of linear equations: \(x+y+z=6\), \(x+2y+3z=14\), and \(x+3y+z=10\). Choose from the following options: a) \(x=1, y=2, z=3\) b) \(x=2, y=3, z=1\) c) \(x=3, y=1, z=2\) d) none of these Answer: (d) none of these

Step-by-step solution

Solve the first equation for x

We will solve the first equation for x by isolating the variable on one side of the equation: \(x+y+z=6\) \(x = 6 - y - z\)

Substitute x in the other two equations

Now we will substitute the expression for x in equations 2 and 3: \(x+2y+3z=14\) \((6-y-z)+2y+3z=14\) \(x+3y+z=10\) \((6-y-z)+3y+z=10\) These two equations will now be in terms of y and z only: \(-y+2z=8 \qquad \text{(Equation A)}\) \(2y-z=-4 \qquad \text{(Equation B)}\)

Solve Equation A for y

Now, we will isolate y in Equation A: \(y=2z - 8\)

Substitute y in Equation B

We will substitute the expression for y in Equation B: \(2(2z-8)-z=-4\)

Solve for z

Now simplify and solve the equation for z: \(4z-16-z=-4\) \(3z=12\) \(z=4\)

Solve for y

Now that we know the value of z, we can substitute this value back into the expression for y: \(y=2(4)-8\) \(y=8-8\) \(y=0\)

Solve for x

Lastly, we will substitute the values of y and z into the expression for x: \(x=6-y-z\) \(x=6-0-4\) \(x=2\) Upon reviewing the given choices, our solution \((x=2, y=0, z=4)\) isn't among the choices. Therefore, the correct answer is:

Answer

(d) none of these

Key concepts

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, matrices, and linear transformations. It's a tool that helps us solve systems of linear equations—like the one presented in our exercise—with ease and efficiency. In linear algebra, equations are often represented as straight lines in space, and solutions to these equations can be visualized as the points where these lines intersect. This aids in understanding the geometric representation of problem-solving.
Linear systems can be expressed in matrix form, which involves rows and columns that encapsulate the coefficients of the variables. This matrix representation not only simplifies solving complex equations but also provides a systematic method to explore solutions. These systems can be further solved using methods such as substitution, elimination, or employing matrix operations like row reduction and inversion.

• Helps visualize and solve systems of equations.
• Provides systematic approaches to equation solving.
• Vital for understanding geometric interpretations.

By mastering these concepts, solving simultaneous equations—where multiple equations are explored at once—becomes a straightforward task.

Equation Solving

Equation solving is the process of finding the values of variables that satisfy given mathematical equations. In the context of simultaneous equations, this involves finding values for each variable that work in all the equations involved. Let's break down how solving these equations typically works:
First, isolate one variable. In our exercise, we first solved one equation for variable as a function of others, like for example isolating, \( x \):

• Express \( x \) with \( x = 6 - y - z \)

This step is crucial because isolating a variable simplifies introducing it into the other equations, thus narrowing the possibilities of values.
Next, substitute this expression into the other equations. This substitution diminishes the system to fewer variables, leading us closer to a solution. You then simplify and solve these new equations. The solution is reaching consistent values for the variables, confirming they satisfy all original equations.

• Isolate variables one at a time.
• Substitute to reduce the number of variables.
• Simplify equations to find solutions.

Through solving, one can arrive at the unique intersection point of the system's solutions.

Systems of Equations

Systems of equations are collections of two or more equations with a common set of variables. Understanding and solving systems is crucial in many areas such as engineering, physics, and finance, because they allow us to model multiple conditions or constraints simultaneously.
In the exercise at hand, we had to deal with a system of three equations:\[\begin{align*}x + y + z &= 6, \x + 2y + 3z &= 14, \x + 3y + z &= 10.\end{align*}\]
Solving such a system requires the implementation of methods that can handle multiple variables. Steps often include isolating variables and substituting them into other equations. This reduces complexity and leads to a manageable system with neat solutions.

• Systems may have a single solution, no solution, or infinitely many solutions.
• Use algebraic methods like substitution or elimination to find solutions.
• Graphically, solutions of such systems can be visualized as points where lines intersect.

Understanding these methods is vital to solving multi-equation problems effectively and correctly.