Problem 103 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} y=-4 x \\ 4 x+y=1 \end{array}\right. $$

Step-by-Step Solution

Answer

The system of equations has no solution.

Step 1: Identify the Equations

The system of equations given is:1) \( y = -4x \)2) \( 4x + y = 1 \)

Step 2: Substitute Equation

Substitute the expression for \( y \) from the first equation into the second equation. This gives:\( 4x + (-4x) = 1 \).

Step 3: Simplify the Equation

Simplify the equation from Step 2:\( 4x - 4x = 0 \), hence the equation simplifies to:\( 0 = 1 \).

Step 4: Solve for x

Since \( 0 = 1 \) is not a true statement, this indicates that there is no solution for \( x \) and subsequently for \( y \) in the system of equations.

Key Concepts

substitution method

The substitution method is one way to solve systems of equations. It works by solving one of the equations for one variable and then substituting this expression into the other equation. This substitution helps us simplify the system to one equation with one variable, making it easier to solve.
For example, in the given system:
\[ y = -4x \] and \[ 4x + y = 1 \],
we can solve the first equation for \( y \). We already have: \[ y = -4x \],
Next, we substitute \( -4x \) for \( y \) in the second equation:
\[ 4x + (-4x) = 1 \]
This simplifies the system to a single equation that is easier to solve.

no solution

Sometimes, you might encounter a system of equations that has no solution. This happens when the equations are contradictory. In our example, after substituting and simplifying, we get: \[ 4x - 4x = 1 \],
which simplifies to \[ 0 = 1 \]. This is not true, so there is no solution.
No matter what \( x \) you choose, you can't make \( 0 = 1 \) true. This indicates the system of equations does not have a solution that works for both equations at the same time.

inconsistent equations

When a system of equations has no solution, it is called inconsistent. Inconsistent equations mean that the lines represented by the equations do not intersect. Instead, they are parallel.
For the given system: \[ y = -4x \] and \[ 4x + y = 1 \],
if we simplify the second equation using the substitution method, we end up with: \[ 0 = 1 \]. This contradiction shows the lines are parallel and never meet.
Therefore, the system is inconsistent and there is no point that satisfies both equations. Always look out for these contradictions when solving systems of equations.