Problem 30 - College Algebra

Question

Solve each system by the addition method. \(\left\\{\begin{array}{l}5 x-6 y+40 \\ 2 y-8-3 x\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = -\frac{56}{19}\) and \(y = -\frac{11}{38}\).

Step 1: Rearrange the Equations

Rearrange the system of equations so that the x terms, y terms, and constants are all vertically aligned in both equations. The system, in standard form \(Ax + By = C\), should look like this: \[\left\{\begin{array}{l}5x - 6y = -40 \ 3x + 2y = 8\end{array}\right.\]

Step 2: Multiply to have Common Coefficients

Before adding the equations together, multiply both sides of each equation by a certain number so that the coefficients of the y variable in both equations will cancel each other out when added. It's necessary to multiply the first equation by 2 and the second equation by 3. The system of equations becomes:\[\left\{\begin{array}{l}10x - 12y = -80 \ 9x + 6y = 24\end{array}\right.\]

Step 3: Add the Equations

Add the two equations together. Because the y variable has been eliminated, you now have an equation that can be solved for the x variable:\[19x = -56\]

Step 4: Solve for x

Solve for x by dividing both sides of the equation by 19. The solution for x is:\[x = -\frac{56}{19}\]

Step 5: Substitute x into the First Equation

Substitute the value of x into the first equation of the initial system, then simplify to solve for y:\[5(-\frac{56}{19}) - 6y = -40\]

Step 6: Solve for y

After simplifying the equation, you can solve for the value of y by isolating y on one side of the equation. The solution for y is:\[y = -\frac{11}{38}\].

Step 7: Check the solution

Substitute \(x = -\frac{56}{19}\) and \(y = -\frac{11}{38}\) back into the original equations to verify that they satisfy both equations.

Key Concepts

Addition Method

The addition method, also referred to as the elimination method, is a popular technique for solving systems of linear equations. This method helps to eliminate one of the variables by adding the equations together, hence producing a simpler equation involving only one of the variables. The key steps in this method include arranging the system of equations in a format where adding them directly cancels one of the variables. This often involves multiplying one or both of the equations by certain numbers so that the coefficients of one variable become equal in magnitude. When you add the two equations together, the target variable is eliminated, simplifying the process of solving for the unknown quantities.

Linear Equations

Linear equations are mathematical expressions that represent straight lines in a Cartesian coordinate system. Typically written in the form of \(Ax + By = C\), they involve constants and variables with no exponents or complex functions such as sine or logarithms. Each term in a linear equation produces a straight line when graphed, and linear equations can be combined in systems that describe multiple constraints or situations. Systems of linear equations allow you to find the point of intersection between the lines, representing the set of values for the variables that satisfy all the equations simultaneously.

Substitution Method

The substitution method is another strategy for solving systems of linear equations. Unlike the addition method, this approach involves solving one of the equations for one variable and then substituting this expression into the other equation. This reduces the number of variables in the second equation, making it easier to solve. The method is beneficial when one of the equations can easily be rearranged to express one variable in terms of the other. Once a solution is found for one variable, substitute back into the original equation to determine the other variable. This method is systematic, often clearer in steps, but can be cumbersome when dealing with equations that result in difficult algebraic manipulations.

Elimination Method

The elimination method, synonymous with the addition method, is best used when you want to eliminate variables by adding or subtracting equations. It involves aligning equations such that the addition of equations leads to the annulment of one variable, streamlining the pathway to finding the solution. This technique typically requires the equations to be adjusted by multiplication to ensure that one of the variables can be easily canceled. After elimination, a simpler equation emerges, with only one variable remaining, which can be solved quickly. The subsequent step involves back substitution to find the other variable, ensuring that both original equations are satisfied.