Problem 10 - Algebra A Combined Function

Question

Solve each system. $$ \left\\{\begin{array}{r} 5 y-7 z=14 \\ 2 x+y+4 z=10 \\ 2 x+6 y-3 z=30 \end{array}\right. $$

Step-by-Step Solution

Answer

There's no solution; the system is inconsistent.

Step 1: Isolate Variables in Equation 1 for y

The given system of equations is: 1. \( 5y - 7z = 14 \) 2. \( 2x + y + 4z = 10 \) 3. \( 2x + 6y - 3z = 30 \). From Equation 1, solve for \( y \): \[ y = \frac{14 + 7z}{5} \].

Step 2: Substitute for y in Equation 2

Substitute \( y = \frac{14 + 7z}{5} \) from Step 1 into Equation 2: \[ 2x + \left(\frac{14 + 7z}{5}\right) + 4z = 10 \]Clear the fraction by multiplying through by 5:\[ 10x + 14 + 7z + 20z = 50 \].

Step 3: Simplify Equation 2

Simplify the equation from Step 2:Combine like terms:\[ 10x + 27z = 36 \] Subtract 14:\[ 10x + 27z = 36 \] (adjusted for simplification error).

Step 4: Substitute for y in Equation 3

Substitute \( y = \frac{14 + 7z}{5} \) into Equation 3: \[ 2x + 6\left(\frac{14 + 7z}{5}\right) - 3z = 30 \]Clear the fractions by multiplying through by 5:\[ 10x + 6(14 + 7z) - 15z = 150 \].

Step 5: Simplify Equation 3

Expand terms from Step 4:\[ 10x + 84 + 42z - 15z = 150 \]Combine like terms:\[ 10x + 27z = 66 \].Rearrange to show second form:\[ 10x + 27z = 66 \] (double check steps as supposed 66) is pruned for realistic steps in current format.

Step 6: Solve for z using Equations from Steps 3 and 5

From the simplified equations in Steps 3 and 5:1. \( 10x + 27z = 36 \)2. \( 10x + 27z = 66 \)This is a contradiction because setting them equal leads nowhere but checking for self-mirror terms uncovers same not held by least squares in algebra.

Step 7: Solve for x given equivalence of Equations is invalid

Recognize inconsistency, additional validation checking at actual function point of resolve exact solutions; otherwise repeat steps for distinct checks.

Key Concepts

Linear Equations

Linear equations are the foundation of this system of equations exercise. A linear equation is one where each term is either a constant or the product of a constant and a single variable. These equations form a straight line when graphed.
Linear equations in a system, like the one we have here, are often represented as:

• 5y - 7z = 14
• 2x + y + 4z = 10
• 2x + 6y - 3z = 30

The goal is to find a common solution that satisfies each linear equation in the system. The equations contain variables, which are the unknowns we want to solve for, and coefficients, which are the numbers multiplying the variables. Solving a system of linear equations often involves finding values for each variable that satisfy all given equations simultaneously.

Variable Isolation

Variable isolation is the process of rearranging an equation so that one of its variables stands alone on one side of the equation. This is an important step in solving systems of equations because it allows for easier substitution into other equations.
In our situation, we isolated the variable \( y \) in the first equation:

• From \( 5y - 7z = 14 \) to \( y = \frac{14 + 7z}{5} \)

By achieving this form, \( y \) can now be replaced by \( \frac{14 + 7z}{5} \) in subsequent equations. Isolation makes future steps of solving a system much more efficient and streamlined. Remember to perform the same operation on both sides of the equation to maintain its balance.

Substitution Method

The substitution method involves using an isolated variable and substituting its equivalent expression into another equation within the system. This step helps in reducing the number of variables, making equations easier to solve.
In our exercise, following the isolation of \( y \), we substitute \( y = \frac{14 + 7z}{5} \) into the second and third equations:

• Into Equation 2: \( 2x + \left(\frac{14 + 7z}{5}\right) + 4z = 10 \)
• Into Equation 3: \( 2x + 6\left(\frac{14 + 7z}{5}\right) - 3z = 30 \)

This method simplifies the complexity of the system by effectively reducing it to more manageable terms. It replaces one variable in terms of known quantities and other variables, progressively narrowing down the solution space.

Equation Simplification

Equation simplification involves combining like terms and performing algebraic operations to condense equations into their simplest forms.
This process makes it easier to compare equations and observe potential solutions or inconsistencies. After substitution, the next step involves simplifying the substituted equations:

• Simplify the substituted version of Equation 2 to \( 10x + 27z = 36 \)
• Simplify the substituted version of Equation 3 to \( 10x + 27z = 66 \)

Sometimes simplification reveals contradictions, as seen with the inconsistent equations \( 10x + 27z = 36 \) and \( 10x + 27z = 66 \). This means there's no common solution satisfying the entire system, highlighting the importance of checking work frequently and ensuring steps are executed correctly. Simplification helps ensure clarity and correctness in solutions, serving as a critical step in solving systems of equations.