Question

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{l}3 x-5 y=2 \\ 2 x+5 y=13\end{array}\right.$$

Short answer

The solution to the system is \(x = 3\) and \(y = 1.4\). The system is consistent as it has at least one solution.

Step-by-step solution

Solve Using Elimination

To solve using elimination, add both equations which will lead to the cancellation of \(y\), resulting: \begin{array} 3x - 5y + 2x + 5y = 2 + 13 \ 5x = 15 \end{array}

Solve for \(x\)

Isolate \(x\) by dividing both sides of the equation by 5: \( x = 15 / 5 = 3.\)

Substitute \(x\)'s Value in the First Equation

Turn to the first equation, substitute \(x = 3\), you get: 3*3 - 5y = 2. Then solve for \(y\), it simplifies to \( -5y = 2-9 = -7.\) So, \(y = -7/-5 = 7/5 \) or \(1.4\).

Check Solution

Plug \(x = 3\) and \(y = 1.4\) into the second equation to check if it's a valid solution: \(2*3 + 5*1.4 = 6 + 7 = 13\) verifies the solution.

Determine the Consistency of the System

Since the substitution satisfies both equations, the system is consistent.

Key concepts

Elimination Method

The elimination method is a popular technique for solving systems of linear equations. It involves manipulating the equations to eliminate one of the variables. This helps in solving for the remaining variable and ultimately finding the solution to the system. The goal is to add or subtract equations such that one variable cancels out.

In our example, we have two equations: - \(3x - 5y = 2\) - \(2x + 5y = 13\)Notice the coefficients of \(y\) are \(-5\) and \(5\). By adding these two equations, \(y\) gets eliminated:

• Add the equations: \( (3x - 5y) + (2x + 5y) = 2 + 13\)
• This simplifies to: \(5x = 15\)

Now, \(y\) has been successfully eliminated, and we are left with a single equation in terms of \(x\) alone.

Thus, the elimination method simplifies the process, making it easier to solve for one variable. Practical, huh?

Consistent and Inconsistent Systems

Once you have solved a system of equations, the next step is to determine if it is consistent or inconsistent.

- **Consistent systems** have at least one solution. This means the equations intersect at a point on the graph. In our example, both equations came together nicely to provide a solution \((x, y) = (3, 1.4)\).- **Inconsistent systems**, on the other hand, do not have any solutions at all. The lines represented by these equations are parallel, meaning they never meet.Checking your solution helps you know the nature of your system: If substituting determined values of \(x\) and \(y\) into either equation satisfies both, your system is consistent.

In our case, plugging \(x = 3\) and \(y = 1.4\) into the second equation \(2x + 5y = 13\) shows correctness, confirming consistency.

Solving Linear Equations

Solving linear equations is at the heart of finding solutions to a system. It means finding values of variables that make the equation true.

Once you use the elimination method to remove one variable, you solve the resulting simple linear equation to find the remaining variable:

• From \(5x = 15\), solving for \(x\) is straightforward: divide both sides by 5 to get \(x = 3\).
• Substitute \(x\) back into one of the original equations to solve for \(y\).
• Using \(3x - 5y = 2\), when \(x = 3\), it becomes \(-5y = 2 - 9 = -7\). Thus, \(y = \frac{7}{5} = 1.4\).

Solving linear equations requires a systematic approach: substitute and validate. Understanding each step can profoundly impact your ability to tackle more complex problems in algebra. Plus, it builds a strong foundation for understanding systems of equations.