Problem 9 - Algebra A Combined Function

Question

Solve each system. $$ \left\\{\begin{aligned} 4 x-y+2 z &=5 \\ 2 y+z &=4 \\ 4 x+y+3 z &=10 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system has no solutions due to a contradiction in the equations.

Step 1: Solve for y in the second equation

The second equation is given by \(2y + z = 4\). We can express \(z\) in terms of \(y\) by rearranging the equation: \(z = 4 - 2y\).

Step 2: Substitute z into the first and third equations

Using the expression for \(z\) from Step 1, substitute \(z = 4 - 2y\) into both the first equation \(4x - y + 2z = 5\) and the third equation \(4x + y + 3z = 10\). This gives us two new equations: \(4x - y + 2(4 - 2y) = 5\) and \(4x + y + 3(4 - 2y) = 10\).

Step 3: Simplify the equations

Simplify the equations from Step 2. The first equation becomes \(4x - y + 8 - 4y = 5\), or \(4x - 5y = -3\). The third equation simplifies to \(4x + y + 12 - 6y = 10\), or \(4x - 5y = -2\).

Step 4: Compare the simplified equations

From Step 3, we have the equations \(4x - 5y = -3\) and \(4x - 5y = -2\). Since both have the same left side but different right sides, there is a contradiction, indicating there are no solutions.

Key Concepts

Substitution Method

The substitution method is a strategic way to solve systems of linear equations. The idea is to solve one of the equations for one variable and then substitute this expression into the other equations. This method simplifies the number of variables, making the problem easier to solve.

• Choose an equation: Pick one equation to solve for one of the variables. Ideally, select an equation where solving for a variable is straightforward.
• Substitute: Replace that variable in the other equations using the expression derived from the first equation.
• Simplify and solve: Work with the reduced system of equations that has fewer variables. Often this leads to a single equation with one unknown that can be easily solved.

In the given exercise, we started by isolating the variable \(z\) in terms of \(y\) using the second equation. This expression for \(z\) was then substituted into the other equations. This substitution is key to simplifying and eventually solving the system.

Inconsistent System

An inconsistent system of linear equations occurs when there is no solution that satisfies all the equations simultaneously. This happens when the equations contradict each other.

• Same left side, different right side: The equations might look similar concerning the variables but have different results. This shows that the equations can't both be true simultaneously.
• Geometrically: If you graph these equations in a coordinate plane, you'll find that the lines or planes do not intersect, meaning there is no common point that satisfies them all.

In our solution, we reached a point where the simplified equations were \(4x - 5y = -3\) and \(4x - 5y = -2\). These equations are inconsistent because they suggest different values for the same expression, indicating there is no possible solution that fits both.

Linear Equations

Linear equations form the basis of solving systems of equations and are represented as straight lines when graphed. Each linear equation in the system represents a straight line on a two-dimensional graph.

• General form: Typically expressed as \(ax + by + cz = d\), where \(a, b, c\), and \(d\) are constants.
• Plane intersections: In three dimensions, these equations represent planes. Solving such a system often involves finding the intersection point, line, or confirming no common geometry exists among the planes.

Understanding linear equations and their properties is crucial, as it forms the backbone for more complex algebraic problems. Our system attempted to find common intersections among these linear equations.

Algebraic Manipulation

Algebraic manipulation involves applying algebraic rules to simplify or transform equations. This is essential in solving systems of equations, especially when using methods like substitution.

• Rearranging terms: Moving terms across the equation to isolate variables or expressions.
• Combining like terms: Gather and simplify expressions that involve similar variables or constants.
• Using distributive property: Apply \(a(b + c) = ab + ac\) to expand or factor expressions in equations.

In the textbook exercise, algebraic manipulation was used throughout the solution process, initially to express \(z\) in terms of \(y\), and then to simplify the resulting equations after substitution. This showcases how movement of terms and combining like terms are fundamental tools in solving systems of equations.