Problem 34 - Algebra Form and Function

Question

Solve the systems of equations in Problems \(34-41\). $$ \left\\{\begin{array}{l} 2 x+5 y=1 \\ 2 y-3 x=8 \end{array}\right. $$

Step-by-Step Solution

Answer

Question: What is the solution of the given system of linear equations: $$ \left\\{\begin{array}{l} 2 x+5 y=1 \\\ 2 y-3 x=8 \end{array}\right. $$ Answer: The solution to the given system of linear equations is (x, y) = (-2, 1).

Step 1: Write down the system of equations

First, write down the given system of linear equations: $$ \left\\{\begin{array}{l} 2 x+5 y=1 \\\ 2 y-3 x=8 \end{array}\right. $$

Step 2: Multiply the equations to make the coefficients of one variable the same

To use the elimination method, we want the coefficients of either x or y to be equal in both equations. In this case, let's choose to make the coefficients of x the same: $$ \left\\{\begin{array}{l} (2)(2 x+5 y)=1(2) \\\ (-3)(2 y-3 x)=8(-3) \end{array}\right. $$ Which results in: $$ \left\\{\begin{array}{l} 4 x+10 y=2 \\\ -6 y+9 x=-24 \end{array}\right. $$

Step 3: Add the equations to eliminate the x variable

Now, add the two equations: $$(4x + 10y) + (-6y + 9x) = 2 +(-24)$$ Which simplifies to: $$13x + 4y = -22$$

Step 4: Solve for one variable

In this case, let's solve for x: $$x = \frac{-22-4y}{13}$$

Step 5: Substitute this expression in one of the original equations

Substitute the expression for x into the first equation of the original system: $$2 \left(\frac{-22-4y}{13}\right) + 5y = 1$$

Step 6: Solve for y

Now, solve for y by multiplying both sides by 13: $$2(-22-4y) + 5(13y) = 1(13)$$ Simplify the equation: $$-44 - 8y + 65y = 13$$ Combine like terms: $$57y - 44 = 13$$ Move -44 to the other side: $$57y = 57$$ Now divide both sides by 57: $$y = 1$$

Step 7: Substitute y back into the expression for x

With the value of y, we can now substitute it back into our expression for x: $$x = \frac{-22-4(1)}{13}$$ Which simplifies to: $$x = \frac{-26}{13}$$ So, x = -2.

Step 8: Write the solution

The solution to the system of equations is: $$(x,y) = (-2,1)$$

Key Concepts

Elimination Method

The elimination method is a popular technique used to solve systems of linear equations. It involves removing one variable so that you can solve for the other. Here’s how it works step by step:

• First, ensure both equations are aligned, with each term present in a similar format. We're dealing with: \[ \begin{aligned} &2x + 5y = 1 \ &2y - 3x = 8 \end{aligned} \]

To eliminate a variable, adjust the equations so that when they are added or subtracted, one set of variables cancels out. In our exercise, the goal was to eliminate the "x" variable.

• This was achieved by multiplying the top equation by 2 and the bottom by 3 (noting the sign change on the second equation to simplify addition): \[ \begin{aligned} &4x + 10y = 2 \ &9x - 6y = -24 \end{aligned} \]
• Notice how these operations make it possible to cancel out the "x" variables by adding these resulting equations.

The beauty of the elimination method is that it transforms the system into a much simpler one-variable equation, which is easier to solve.

Linear Equations

Linear equations are algebraic equations in which the highest power of the variable(s) is one. They graph as straight lines on a coordinate plane.

• In the given exercise, we are working with a system of linear equations: \[ \begin{aligned} &2x + 5y = 1 \ &2y - 3x = 8 \end{aligned} \]

This means each equation can represent a line in a two-dimensional space. Where these lines intersect, you find the solution to the system.

• The simplicity of linear equations permits methods like substitution and elimination to effectively solve them.
• Equations are usually balanced using operations like addition, subtraction, multiplication, or division — just as seen in the elimination method process.

The goal is to find variable values that satisfy all equations in the system simultaneously. This implies that at the point of intersection, those variable values are true for each equation.

Solution Verification

Solution verification is an important step to ensure the accuracy of your solution to a system of equations. Once you have determined a potential solution, such as \((x, y) = (-2, 1)\), it’s essential to substitute these values back into the original equations.
To verify our solution:

• Take the first equation: \(2(-2) + 5(1) = 1\). Computation: \(-4 + 5 = 1\), which holds true.
• Check the second equation: \(2(1) - 3(-2) = 8\). Computation: \(2 + 6 = 8\), confirming truth.

If both originally given equations are satisfied, the solution \((x, y) = (-2, 1)\) is verified.
This process helps to catch any mistakes made during calculations and ensure the solution fits all given conditions of the problem. Remember, thorough verification contributes greatly to reliable results.