Problem 92 - College Algebra with Modeling and Visualization

Question

Solve the system, if possible. $$ \begin{array}{rr} -5 x+3 y= & -36 \\ 4 x-5 y= & 34 \end{array} $$

Step-by-Step Solution

Answer

The solution is \((6, -2)\) for the system.

Step 1: Choose a method for solving

For this system, we'll use the method of substitution. We'll solve the first equation for one variable and then substitute into the second equation.

Step 2: Solve for y in the first equation

The first equation is \(-5x + 3y = -36\).Let's solve for \(y\):Add \(5x\) to both sides:\(3y = 5x - 36\).Divide every term by 3:\(y = \frac{5x - 36}{3}\).

Step 3: Substitute y in the second equation

Substitute \(y = \frac{5x - 36}{3}\) into the second equation:The second equation is \(4x - 5y = 34\).Substitute the expression for \(y\):\(4x - 5(\frac{5x - 36}{3}) = 34\).

Step 4: Solve the equation for x

First, simplify \(-5(\frac{5x - 36}{3})\):\(-\frac{25x - 180}{3}\).Then the equation becomes:\[4x - \frac{25x - 180}{3} = 34\].Multiply every term by 3 to clear the fraction:\(12x - 25x + 180 = 102\).Simplify:\(-13x = -78\).Divide both sides by -13:\(x = 6\).

Step 5: Substitute x back to find y

Now that we have \(x = 6\), substitute into \(y = \frac{5x - 36}{3}\).\(y = \frac{5(6) - 36}{3}\).Calculate:\(y = \frac{30 - 36}{3} = \frac{-6}{3} = -2\).

Step 6: Verify the solution

Let's verify the solution by checking both equations:For \(-5x + 3y = -36\):\(-5(6) + 3(-2) = -30 - 6 = -36\). Correct.For \(4x - 5y = 34\):\(4(6) - 5(-2) = 24 + 10 = 34\).Correct. Therefore, the solution is verified.

Key Concepts

Substitution Method

The substitution method is a straightforward technique for solving systems of equations involving two equations with two variables. This method involves finding an expression for one variable in terms of the other from one equation, and then substituting it into the other equation. By doing this, we essentially reduce the system to a single equation with one variable, making it easier to solve.

In our example, we first solved for \( y \) from the first equation \(-5x + 3y = -36\). By adding \( 5x \) to both sides and then dividing by 3, we expressed \( y \) in terms of \( x \). This expression, \( y = \frac{5x - 36}{3} \), was then substituted into the second equation \( 4x - 5y = 34 \).

Substitution is particularly useful because it organizes the problem systematically:

• Step 1: Isolate one variable.
• Step 2: Substitute this into the other equation.
• Step 3: Solve for the remaining variable.

These steps simplify complex problems, making substitution beneficial for readily solving systems of equations.

Algebraic Solutions

Algebra involves using mathematical expressions and principles to solve equations and find unknown values. When we solve equations algebraically, we focus on manipulating the expressions to isolate and determine the variables.

In this example, once we obtained \( x \) using the substitution method, we returned to algebraic techniques to find \( y \). We substituted \( x = 6 \) back into the expression \( y = \frac{5x - 36}{3} \). This substitution allowed us to solve for the missing variable \( y \), resulting in \( y = -2 \).

The algebraic process is all about maintaining equality and logical steps:

• Perform operations equally on both sides of an equation.
• Use inverse operations to isolate variables.
• Continuously check for simplification opportunities.

By building on basic arithmetic and algebra principles, solving these equations becomes more manageable.

Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. They graph as straight lines and can easily represent relationships between variables.

In our system of equations, both \(-5x + 3y = -36\) and \(4x - 5y = 34\) are linear. They represent two lines in a 2-dimensional space, and the solutions to these equations are points of intersection of these lines.

Key characteristics of linear equations include:

• They can be written in the form \(ax + by = c\).
• The highest power of any variable is 1.
• They produce straight lines when graphed.

Understanding linear equations helps to visualize problems, as the solution to the system represents where the two lines meet.

Verification of Solutions

Verification is a crucial final step in solving systems of equations. It ensures that the solutions obtained are correct and satisfy both original equations.

During verification, we substitute the solution pair \((x, y) = (6, -2)\) back into the original equations:
1. For the equation \(-5x + 3y = -36\), substituting \(x = 6\) and \(y = -2\) gives \(-5(6) + 3(-2) = -36\), which is true.
2. For \(4x - 5y = 34\), substituting the same values results in \(4(6) - 5(-2) = 34\), also true.

This process confirms that our determined values for \(x\) and \(y\) satisfy both equations, verifying our solution. Verification reaffirms that:

• The calculations were performed correctly.
• The solution satisfies all criteria of the system.
• The answer is logically consistent with the problem's constraints.

Completing these steps builds confidence in the solution.