Question

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{aligned} 4 x-3 y &=11 \\\\-6 x+3 y &=3 \end{aligned}\right.$$

Short answer

The system is consistent and the solution is \(x = -7, y = -13\).

Step-by-step solution

Add the equations

Add the two equations together. This will eliminate \(y\) because \( -3y + 3y = 0\). The resulting equation will be: \[4x - 6x = 11 + 3\] This simplifies to: \[ -2x = 14. \]

Solve for the remaining variable

With only one variable left in the equation, solve for \(x\). Because the equation is \(-2x = 14\), divide both sides by \(-2\) to find \(x\). This gives: \[x = -7\].

Substitute x value back into one of the original equations

Substitute \(x = -7\) back into one of the original equations in order to solve for \(y\). By substituting into the first equation: \[4(-7) - 3y = 11\] which simplifies to: \[ -28 - 3y = 11\]. Solving for \(y\) we get \( y = -13\).

Analyze the system

After solving for \(x\) and \(y\), the system is consistent because a solution exists.

Key concepts

Elimination Method

The elimination method is a powerful technique for solving systems of linear equations. This method focuses on removing one variable, making it easier to solve for the other. In our given system, the elimination method is applied because the coefficients of the variable \(y\) are opposites in the two equations: \(-3y\) and \(3y\). By adding these equations together, \(y\) is eliminated, simplifying the system to a single equation in terms of \(x\).

This technique helps simplify the system and is particularly useful when variables have coefficients that can easily be canceled out. The key is to strategically add or subtract equations, focusing on creating zeros for one of the variables.

• Align terms vertically.
• Add or subtract equations to eliminate a variable.
• Once a variable is eliminated, solve for the remaining variable.

Using elimination can often transform complex systems into simpler, more manageable problems.

Consistent and Inconsistent Systems

When solving systems of equations, it's essential to determine whether the system is consistent or inconsistent. A consistent system has at least one solution. It means the lines represented by the equations intersect at some point. In contrast, an inconsistent system has no solutions; the lines are parallel and never meet.

In our exercise, after solving for \(x\) and \(y\), we find specific values for both variables. This result indicates that the system is consistent since it has a point of intersection. It’s a crucial step to confirm the existence of a solution as it explains the relationship between the two equations geometrically: their graphs intersect at a single point.

Algebraic Solutions

Algebraic solutions involve using algebraic operations to find the values of variables. In the context of our exercise, once \(y\) was eliminated through the elimination method, we were left with the equation \(-2x = 14\). Solving this provided the value of \(x\).

After determining \(x\), substitution helps find \(y\). This algebraic process relies strictly on mathematical operations like addition, subtraction, multiplication, and division to reach a final solution.

Steps for algebraic solutions include:

• Simplifying equations as much as possible.
• Isolating variables using inverse operations.
• Substituting known values back into original equations to find other unknowns.

This approach is systematic and straightforward, utilizing the properties of equality and operations to find a solution.

Equation Solving

Equation solving is the backbone of finding solutions in algebra. It encompasses the process of manipulating equations to isolate variables and determine their values. In our system, starting with two equations, the elimination method made it possible to simplify and solve for \(x\) first.

Once \(x\) was found to be \(-7\), we substituted it back into one of the original equations to solve for \(y\). This step is crucial as it ensures that the solution satisfies both of the original equations. This process is based on the foundational principle that properties of equality can be used to maintain the balance of equations while isolating the variables.

Equation solving involves:

• Using additive and multiplicative inverses to simplify equations.
• Checking solutions in the original equations to verify accuracy.
• Iterative steps to ensure all variables are accounted for.

These principles ensure that the solutions are both precise and valid.