Problem 7 - Algebra 1: Concepts and Skills

Question

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. $$\begin{aligned} &-x+y=7\\\ &2 x-2 y=-18 \end{aligned}$$

Step-by-Step Solution

Answer

The system of equations has no solutions.

Step 1: Prepare the Equation for Combination

Multiply the first equation by 2 in order to match the coefficients with the second equation: \(2(-x+y) = 2*7\), which simplifies to \(-2x + 2y = 14\). Now the system of equations is: \(-2x+2y=14\) and \(2x-2y=-18\).

Step 2: Use the Combination Method to Solve

Add the two equations to cancel out the y-terms: \(((-2x + 2y) + (2x - 2y)) = (14-18)\) which simplifies to \(0 = -4\).

Step 3: Check for Inconsistencies

Checking the result 0 = -4, we see a clear inconsistency as zero cannot be equal to -4. The result implies that this system of equation has no solutions.

Key Concepts

Substitution Method

The substitution method is a powerful tool used to solve systems of linear equations. It is especially useful when equations are easy to manipulate. This method involves solving one of the equations for one variable and then substituting that expression into the other equation. This process effectively reduces the number of variables, allowing you to solve for one unknown.

Let's see how it works:

• Solve one equation for one of the variables in terms of the other variable. For instance, if you have the equation \(x + y = 5\), you could solve for \(y\) to get \(y = 5 - x\).
• Substitute this expression into the other equation. This step eliminates the variable you solved for earlier, reducing the system to one equation with one unknown.
• Solve the new equation. Find the value of the remaining unknown.
• Substitute the found value back into the expression obtained in the first step to find the value of the other variable.

This method is best suited for systems where it is easy to isolate a variable or in cases involving less complex numbers.

Linear Combinations

The linear combination method, also known as the elimination method, is another strategy to solve systems of linear equations. This method focuses on adding or subtracting equations to eliminate one of the variables, making the system simpler to solve.

To perform linear combinations:

• Align the equations, ensuring one variable has matching or opposite coefficients.
• Multiply one or both of the equations by a necessary factor to make the coefficients of a variable equal or opposite.
• Add or subtract the equations to eliminate that particular variable. You will get an equation with only one variable, which you can easily solve.
• Use the value obtained to substitute back into one of the original equations to find the other variable.

In this exercise, the linear combination method revealed an inconsistency, indicating no solution. This result occurs because the equations represent parallel lines that never intersect.

Inconsistent Systems

An inconsistent system of linear equations is one that has no solution. This happens when the equations in the system represent parallel lines when plotted on a graph. Parallel lines never meet, hence there is no point that satisfies all equations simultaneously.

You can identify an inconsistent system by:

• Performing operations to eliminate a variable and encountering a false statement, such as \(0 = -4\), indicating a contradiction.
• Recognizing that the equations are equivalent in slope but different in intercept, implying parallelism.

In the step-by-step solution provided earlier, after using the linear combination method, it was evident with the false equality that the system was inconsistent.
Understanding whether a system is inconsistent helps in identifying the nature of the equations involved and allows decision making on the need for alternate methods or verifying initial conditions.