Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{l} 2 x+4 y+z=-4 \\ 2 x-4 y+6 z=13 \\ 4 x-2 y+z=6 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution for the system of equations is x = -1, y = 1, and z = -4.

Step 1: Simplify the Equations

Since all equations have common factors, we can simplify them by dividing the whole equation by their common factors: \(x + 2y + 0.5z = -2\), \(x - 2y + 3z = 6.5\), and \(2x - y + 0.5z = 3\)

Step 2: Rearrange the equations

Rearrange the first and third equations in terms of x: \(x = -2 - 2y - 0.5z\) and \(2x = 3 + y - 0.5z\)

Step 3: Elimination of x

Substitute equation (1) into equation (2) and simplify to solve for y: \(2(-2 - 2y - 0.5z) = 3 + y - 0.5z\). This gives y = 1.

Step 4: Solve for z

Substitute y = 1 into the first equation and solve for z: \(x + 2(1) + 0.5z = -2\). This gives z = -4.

Step 5: Solve for x

Substitute y = 1 and z = -4 into the first equation and solve for x. This gives x = -1.

Step 6: Check the solution

Substitute x = -1, y = 1, and z = -4 into the original equations to verify the solution.

Key Concepts

Elimination Method

The elimination method is a popular technique used to solve systems of linear equations. It involves manipulating the equations to eliminate one of the variables, simplifying the system until it is solvable. Our goal is to reduce it to a simple one-variable equation. To achieve this, we perform operations to both sides of the equations, such as addition, subtraction, multiplication, or division, aimed at canceling out one variable.

In the original problem, we have three equations with three variables. At Step 3, the elimination method was applied to eliminate the variable \(x\). By substituting the expression for \(x\) derived from one equation into another, we isolate \(y\). Here, the equations are multiplied, added, or subtracted until one variable is eliminated, allowing us to focus on the remaining variables.

Key steps include:

• Align the equations for easy manipulation.
• Perform operations to eliminate one variable.
• Solve the resulting simpler system.

Applying this method can solve otherwise complex systems efficiently.

Substitution Method

The substitution method provides another approach to solving systems of linear equations by directly substituting one variable's expression into another equation. This method is practical when one of the equations is already solved for a single variable or can be easily manipulated to isolate that variable.

During the given exercise, the substitution method was effectively used after simplifying the systems and isolating \(x\) in terms of \(y\) and \(z\). In Step 2, the equation is rewritten to express \(x\) using the other variables. This new expression, \(x = -2 - 2y - 0.5z\), is then substituted into the second adjusted equation, transforming the system into one busy with fewer variables.

Essential steps:

• Isolate one variable in one of the equations.
• Substitute this expression into the other equations.
• Solve the new equation.

This process systematically reduces complexity, enabling us to solve each variable step-by-step.

Solution Verification

Verification is a critical step in solving systems of equations to ensure that the solution is correct and consistent with all original equations. It involves substituting the found solutions back into the original equations.

In the final step of the exercise, \(x = -1\), \(y = 1\), and \(z = -4\) were substituted back into the original set of equations to check for any discrepancies. This process confirms the accuracy of the solutions, helping catch any errors made during calculation or simplification.

Verification procedure includes:

• Substitute each variable into all original equations.
• Confirm that each equation holds true.
• If correct, the system is solved; if not, re-evaluate steps.

Thus, solution verification acts as a reliable confirmation step ensuring the integrity of results derived from complex computations.