Problem 41 - Algebra and Trigonometry

Question

\(21-48=\) Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example \(6 .\) $$ \left\\{\begin{array}{l}{8 s-3 t=-3} \\ {5 s-2 t=-1}\end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system is \((-3, -7)\).

Step 1: Identify the equations

We have the following system of equations:1. \(8s - 3t = -3\)2. \(5s - 2t = -1\)Our goal is to find the values of \(s\) and \(t\) that satisfy both equations simultaneously.

Step 2: Make coefficients of one variable the same

To eliminate one of the variables, we can multiply the equations by suitable factors to make the coefficients of \(t\) equal. Let's multiply the first equation by 2 and the second equation by 3, which gives us:1. \(16s - 6t = -6\)2. \(15s - 6t = -3\)

Step 3: Subtract the equations

Subtract the second equation from the first to eliminate the \(t\) variable:\[(16s - 6t) - (15s - 6t) = -6 - (-3)\]This simplifies to:\[s = -3\]

Step 4: Substitute back to find the other variable

Substitute \(s = -3\) into one of the original equations to find \(t\). Let's use the first equation:\[8(-3) - 3t = -3\]Solve for \(t\):\[-24 - 3t = -3\]Add 24 to both sides:\[-3t = 21\]Divide by -3:\[t = -7\]

Step 5: State the solution

The solution to the system of equations is \(s = -3\) and \(t = -7\). Therefore, the ordered pair solution is \((-3, -7)\).

Key Concepts

Simultaneous Equations

Simultaneous equations involve finding a set of values for variables that satisfy multiple equations at once. In our problem, we are dealing with two equations: \(8s - 3t = -3\) and \(5s - 2t = -1\). To solve these equations simultaneously, we must find values of \(s\) and \(t\), which fit both equations.

Simultaneous equations can represent different relationships, such as lines on a graph. The point where these two lines intersect is the solution that satisfies both equations.

• If the lines intersect at one point, there is a single unique solution.
• If they are parallel, there is no solution.
• If they are identical lines, there are infinitely many solutions.

By using methods like substitution, elimination, or graphing, simultaneous equations can be solved effectively. In this solution, we are particularly using the elimination method.

Elimination Method

The elimination method is a strategic way to solve simultaneous equations. It focuses on removing one of the variables so that the other can be solved more straightforwardly. In our current problem, we use this method by adjusting the coefficients of \(t\) in both equations.

To start, multiply the first equation by 2 and the second equation by 3. This makes the coefficients of \(t\) in both equations equal:
- Equation 1 becomes \(16s - 6t = -6\).
- Equation 2 becomes \(15s - 6t = -3\).

After aligning the coefficients, subtract one equation from the other to eliminate \(t\):

• \((16s - 6t) - (15s - 6t) = -6 - (-3)\)
• This simplifies to \(s = -3\)

Once \(s\) is found, substitute it back into one of the original equations to solve for \(t\), verifying the solution. This method provides a systematic approach and is particularly useful for equations with straightforward coefficients.

Ordered Pair Solution

An ordered pair is a way of displaying the solution to a system of equations as coordinates \((x, y)\). In the context of our problem, it represents the solution \((s, t)\) that satisfies both equations simultaneously.

For our specific solution, we found that \(s = -3\) and \(t = -7\). Therefore, the ordered pair that solves the system of equations is \((-3, -7)\). This pair demonstrates the point of intersection if the system of equations were to be graphed.

Ordered pairs are essential because they provide a visual representation of the solution on a coordinate graph, helping in understanding how different equations relate to each other. Knowing how to interpret these pairs and their relationship in a graph is a fundamental skill in algebra.