Problem 164 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 7 x-9 y=16 \\ -21 x+27 y=-24 \end{array}\right. $$

Step-by-Step Solution

Answer

There is no solution to the system of equations.

Step 1 - Solve for \(x\) in the first equation

Take the first equation of the system, which is \(7x - 9y = 16\). Solve for \(x\): \[7x = 16 + 9y\] \[x = \frac{16 + 9y}{7}\].

Step 2 - Substitute \(x\) into the second equation

Substitute \(x = \frac{16 + 9y}{7}\) into the second equation \(-21x + 27y = -24\). The new equation becomes: \[-21\left(\frac{16 + 9y}{7}\right) + 27y = -24\].

Step 3 - Simplify and solve for \(y\)

Simplify the equation: \[-21 \left(\frac{16 + 9y}{7}\right) = -48 - 27y \] \[ -3(16 + 9y) + 27y = -24 \] \[ -48 - 27y + 27y = -24 \] \[-48 = -24 \]. This is a contradiction, verifying that there is no solution.

Key Concepts

The Substitution Method

The substitution method is an effective way to solve systems of equations. You begin by solving one of the equations for one variable in terms of the other. Then, substitute this expression into the second equation.
This substitution converts the system into a single equation with one variable.
Let's break it down further:

• First, isolate one of the variables (either x or y) in one of the given equations.
• Substitute this variable into the other equation.
• Solve the resulting single-variable equation to find the value of one variable.
• Finally, substitute this value back into the expression from the first step to find the other variable.

In our example, we first solved for x in terms of y from the first equation, then substituted this solution into the second equation. This method is especially useful when the equations given are easier to manipulate and solve algebraically.

Solving Equations

Solving equations is the crux of mathematics, involving finding the value of the unknown variables that make the equation true.
When working with systems of equations, the goal is to find a solution pair \((x, y)\) that satisfies both equations simultaneously.
In our given exercise, follow these steps to solve the equations:

• Write the system of equations clearly.
• Solve one equation for a single variable.
• Substitute this variable into the other equation.
• Simplify the resulting equation.
• Solve for the remaining variable.

After getting the value of one variable, substitute it back into one of the original equations to find the value of the other variable.
This approach efficiently breaks down a multi-variable system into manageable single-variable problems. In the worked example, substitution allowed handling one equation at a time, helping to understand contradictions or mismatches early on.

Algebraic Contradictions

While solving systems of equations, sometimes you may encounter contradictions. An algebraic contradiction occurs when your simplification results in a false statement, like \(-48 = -24\).
This indicates that there are no values for the variables that can simultaneously satisfy both equations.
Such cases are referred to as 'inconsistent systems.' Here’s how you typically identify an algebraic contradiction:

• Follow your solving routine until you notice an absurd statement like \(-48 = -24\).
• Realize that such a statement signifies no possible solutions exist.

Contradictions often arise when the equations essentially describe parallel lines in two-dimensional space, meaning they never intersect.
Thus, no single solution exists that satisfies both equations at once. This insight is key in identifying and understanding the nature of the equations you are dealing with.