Problem 37 - Elementary Algebra

Question

Solve each system by using either the substitution or the elimination-by- addition method, whichever seems more appropriate. $$\left(\begin{array}{l}5(x+1)-(y+3)=-6 \\\ 2(x-2)+3(y-1)=0\end{array}\right)$$

Step-by-Step Solution

Answer

Solution is \(x = -1\), \(y = 3\).

Step 1: Simplify the Equations

Begin by simplifying each equation in the system.\For the first equation: \[5(x + 1) - (y + 3) = -6\]\[5x + 5 - y - 3 = -6\]\[5x - y + 2 = -6\] This simplifies to:\[5x - y = -8\]For the second equation: \[2(x - 2) + 3(y - 1) = 0\]\[2x - 4 + 3y - 3 = 0\]\[2x + 3y = 7\]

Step 2: Choose a Method

Decide between substitution and elimination. In this case, elimination by addition seems most appropriate because the coefficients of \(y\) are easily manageable.

Step 3: Eliminate a Variable

We'll eliminate \(y\) by adding multiples of the equations. Multiply the first equation by 3 to make the coefficients of \(y\) opposites:\[3(5x - y) = 3(-8)\]Which gives:\[15x - 3y = -24\]Now add this to the second equation:\[15x - 3y = -24\]\[+ \; \;2x + 3y = 7\]------------------\[17x = -17\]

Step 4: Solve for x

Solve the equation from Step 3 for \(x\):\[17x = -17\] Divide by 17:\[x = -1\]

Step 5: Substitute to find y

Use the value of \(x\) in either equation to find \(y\). Let's use the first simplified equation:\[5x - y = -8\]Substitute \(x = -1\):\[5(-1) - y = -8\]\[-5 - y = -8\]Add 5:\[-y = -3\]Multiply by -1:\[y = 3\]

Step 6: Verify the Solution

Substitute \(x = -1\) and \(y = 3\) back into the original system to verify:For the first equation:\[5(-1 + 1) - (3 + 3) = -6\]\[0 - 6 = -6\] Verified.For the second equation:\[2(-1 - 2) + 3(3 - 1) = 0\]\[0 = 0\] Verified. The solution \(x = -1\), \(y = 3\) is correct.

Key Concepts

Substitution Method

The substitution method is a strategy for solving systems of linear equations where you solve one equation for one variable and then substitute that expression into the other equation. This can be especially useful when the system of equations includes straightforward expressions that can be easily isolated.
Here’s a streamlined way to use this method:

• Solve one of the equations for one variable in terms of another. For instance, from the equation \(y = 2x + 3\), isolate \(y\).
• Plug the expression from step one into the other equation. Replace the solved variable with its equivalent algebraic expression in the remaining equation.
• Simplify and solve the resulting one-variable equation.
• Substitute back to find the second variable.

This method is highly beneficial when one of the equations is already solved for a specific variable or can be easily manipulated for substitution.

Elimination Method

The elimination method, sometimes known as the addition method, involves adding or subtracting equations to eliminate one of the variables, allowing you to solve for the other one directly.
Here's how you can efficiently implement this method:

• Look at both equations and decide which variable to eliminate based on easiest coefficients to align through multiplication or simple addition/subtraction.
• Adjust the coefficients by multiplying the entire equations by necessary constants so that they result in opposite coefficients for one of the variables.
• Add or subtract the equations to cancel out one variable.
• Solve for the remaining variable using the simplified equation.
• Substitute the obtained value back into one of the original equations to find the other variable.

This method was particularly appropriate for our original problem because it allowed us to remove \(y\) from the equations smoothly by making their coefficients opposite, leading directly to a solution for \(x\).

Algebraic Simplification

Algebraic simplification involves reducing equations through basic mathematical processes to make them cleaner and easier to solve. It typically includes combining like terms and performing arithmetic operations to present equations in a simpler form.
Here's why it's useful:

• Simplification makes equations more manageable and often paves the way for easier application of techniques like substitution or elimination.
• It ensures equations are consistent in format, helping especially with complex systems involving fractions or multiple terms.
• By reducing clutter, you decrease opportunities for calculation errors.

In our problem, algebraic simplification was a first step—necessary to transform the original equations into a form where employing the elimination method became a clear choice. By simplifying, we reduced the potential for mistakes during arithmetic steps and set the stage for the elimination smoothly.