Problem 17 - Algebra A Combined Function

Question

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} 4 x+2 y=5 \\ -2 x=y+4 \end{array}\right. $$

Step-by-Step Solution

Answer

No solution, the lines are parallel.

Step 1: Solve for y in the second equation

First, take the second equation \(-2x = y + 4\)and solve for \(y\). Start by isolating \(y\) by subtracting 4 from both sides:\(-2x - 4 = y\).

Step 2: Substitute y in the first equation

Now substitute \(y = -2x - 4\) from the second equation into the first equation \(4x + 2y = 5\). This gives:\[4x + 2(-2x - 4) = 5\].

Step 3: Simplify and Solve for x

Simplify the equation:\(4x - 4x - 8 = 5\). This simplifies to \(-8 = 5\),which indicates that there was an error, suggesting that the lines are parallel and don't intersect.

Step 4: Confirm no solution

The above simplification, \(-8 = 5\), is false, confirming that the system of equations is inconsistent. The lines represented by the equations do not intersect, so there is no solution.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of linear equations. Essentially, this method involves solving one of the equations for one variable and then substituting this expression into the other equation.
This process helps to eliminate one of the variables, making it easier to solve for the other.

• Start by solving one of the equations for one of its variables.
• Substitute this expression into the other equation.
• Solve the resulting equation for the remaining variable.
• Substitute back if necessary to find the other variable.

In the given problem, we first solve the second equation (-2x = y + 4) for y, which gives us y = -2x - 4. By substituting this expression for y into the first equation, we aim to find the value of x. If consistent, this approach yields the solution; however, you might end up with a non-viable simplification, which hints at further investigation about the nature of the solution.

Solving Equations

Solving equations is about finding values of variables that satisfy given equations. For linear equations, this usually involves finding particular values for the variables that make both sides of the equation equal.
The key steps include combining like terms and simplifying the expression.

• Start with simplifying both equations separately if necessary.
• Combine like terms so you can isolate the variable.
• If substitution reveals a contradiction (like -8 = 5), check if there is an inherent error or if it suggests an inconsistency.

When using the substitution method in our example, after simplification, the equation turns into the false statement -8 = 5. This is an impossibility in standard algebra, indicating inconsistency or inaccuracy in expected solutions.

Inconsistent System

An inconsistent system of equations occurs when there are no solutions that satisfy all equations simultaneously. This usually happens when the system of equations describes parallel lines.
The algebraic outcome reflects a contradiction.

• When equations have the same slope but different intercepts, they represent parallel lines.
• Inconsistent equations often lead to a false statement on attempting to solve them (like 0 = 1).
• Graphically, the lines do not meet, leading to no possible intersection points and hence, no solution.

In our case, the contradiction -8 = 5 indicates that the equations do not share a common point of solution, thus confirming the inconsistency of the system.

Parallel Lines

Parallel lines in a system of equations imply that the lines run in the same direction and never intersect. This scenario typically results in an inconsistent system of equations.
The absence of solutions is confirmed by the algebraic simplification resulting in an impossibility.

• Lines are parallel when their slopes are equal.
• A different y-intercept means they do not overlap.
• This is visually evident when plotting the equations on a graph.

In our exercise, the equations represented by 4x + 2y = 5 and -2x = y + 4 yield parallel lines due to the equal slopes but differing intercepts, leading to no intersection and confirming an infinite separation between lines.