Problem 16 - Algebra Form and Function

Question

Solve the systems of equations. $$ \left\\{\begin{array}{r} 7 x+5 y=-1 \\ 11 x+8 y=-1 \end{array}\right. $$

Step-by-Step Solution

Answer

Answer: The values that satisfy the given system of linear equations are x = 3 and y = -22/5.

Step 1: Set up the equations for elimination

To eliminate one variable, we will first make the coefficients of either x or y the same in both equations. In this case, we will make the coefficients of y the same by multiplying the first equation by 8 and the second equation by 5. $$ \left\{ \begin{array}{r} (8)(7 x + 5 y) = (8)(-1) \\ (5)(11 x + 8 y) = (5)(-1) \end{array} \right. $$ Now simplify. $$ \left\{ \begin{array}{r} 56 x + 40 y = -8 \\ 55 x + 40 y = -5 \end{array} \right. $$

Step 2: Subtract the equations to eliminate y

Now, subtract the second equation from the first equation to eliminate y. \((56x+40y)- (55x+40y)= -8-(-5)\) Simplify the equation. \(x=3\)

Step 3: Substitute the value of x into one of the original equations

Now that we have the value of x, we can substitute it into one of the original equations to find the value of y. We will use the first equation for this: \(7x+5y=-1\) Substitute x = 3: \(7(3)+5y=-1\)

Step 4: Solve for y

Now, we solve for y: $7(3)+5y=-1\\ 21+5y=-1\\ 5y=-1-21\\ 5y=-22$ To find y, divide by 5: \(y= \dfrac{-22}{5}\)

Step 5: State the solution

The solution to the system of equations is: $$ \left\{ \begin{array}{r} x = 3 \\ y = -\dfrac{22}{5} \end{array} \right. $$

Key Concepts

Understanding the Elimination Method

The **elimination method** is a powerful tool to solve systems of linear equations. This technique involves adding or subtracting equations to eliminate one of the variables, making it easier to solve the system. Here's how it works:

• First, modify the equations so that the coefficients of one of the variables are the same. This could mean multiplying one or both equations by constants. In the example, we adjusted the equations so that the coefficients of \( y \) were both 40.
• Once the coefficients are the same, simply add or subtract the equations from each other to eliminate one of the variables. This leaves you with a single equation with one variable.

The **key advantage** of this method is its straightforward approach, especially when the system involves simple numbers. It is particularly effective when you can see an easy way to make the coefficients align, which then allows you to quickly find the value of the other variable.

Exploring the Substitution Method

The **substitution method** is another technique used to solve systems of linear equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation. Here's a step-by-step breakdown:

• Choose one of the equations and solve it for one variable. In our solution, we could also have used substitution by rearranging the first equation, \( 7x + 5y = -1 \), to express \( y \) in terms of \( x \).
• Substitute this expression into the other equation. This gives you a single equation with one variable, making it simpler to solve.
• Once you have the value of this variable, substitute it back into the equation you first solved to find the other variable.

The substitution method is great when one of the variables is already isolated or can be easily isolated. It is a go-to strategy for tackling systems where one equation is simpler or when dealing with complex fractions.

Basics of Linear Equations

**Linear equations** are the building blocks of algebra and can often be recognized by **their form,** which is usually \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. These equations represent straight lines when graphed on a coordinate plane.Here are some important points about linear equations:

• **Consistency and Dependency**: Systems of linear equations can be consistent (at least one solution) or inconsistent (no solutions). A consistent system can be independent with exactly one solution or dependent with infinitely many solutions.
• **Graphical Interpretation**: Each equation corresponds to a line in a plane. The solution to the system is where the lines intersect (if they do). Two lines that are parallel correspond to an inconsistent system, while the same line represented by different equations indicates a dependent system.

linear equations **form the basis** for more advanced algebraic concepts. Mastery of these can greatly simplify understanding more complex mathematical scenarios. Whether using elimination, substitution, or any other method, solving these equations is a fundamental skill in mathematics.