Problem 39 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 5 x+2 y &=4 y+9 \\ y &=x-3 \end{aligned}\right.$$

Step-by-Step Solution

Answer

x = 1, y = -2

Step 1: Simplify the First Equation

Start by simplifying the first equation. Move all the terms involving y to one side:

Step 2: Rearrange the First Equation

Subtract 2y from both sides: 5x + 2y - 2y = 4y + 9 - 2y This simplifies to: 5x = 2y + 9

Step 3: Substitute the Second Equation

Substitute the expression for y (y = x - 3) from the second equation into the simplified first equation:5x = 2(x - 3) + 9

Step 4: Solve for x

Expand and simplify: 5x = 2x - 6 + 9Combine like terms: 5x = 2x + 3Subtract 2x from both sides: 5x - 2x = 3This results in: 3x = 3Divide both sides by 3: x = 1

Step 5: Solve for y

Use the value of x found in Step 4 (x = 1) and substitute it back into the second equation y = x - 3 to find y: y = 1 - 3 This simplifies to: y = -2

Step 6: Verify the Solution

Substitute x = 1 and y = -2 back into the original system of equations to verify the solution: 5(1) + 2(-2) = 4(-2) + 9This simplifies to: 5 - 4 = -8 + 9 which results in: 1 = 1This confirms that the values are correct.

Key Concepts

Substitution Method

The substitution method is a handy technique for solving systems of equations. It involves solving one equation for one variable and then substituting that value into the other equation. This way, we essentially reduce the system to a single equation with one variable. Let’s break this down:

First, we start by isolating one of the variables in one of the equations. In our exercise, we are given two equations:

• 5x + 2y = 4y + 9
• y = x - 3

Here, the second equation is already solved for y. That makes it easier because we can directly use this expression for y (which is y = x - 3) and substitute it into the first equation. This substitution simplifies the process significantly by reducing the number of variables.

Once we substitute y with x - 3 in the first equation, we can solve for x easily. After finding the value of x, we can then substitute it back into the second equation to find the value of y. This step-by-step approach ensures that we systematically solve for both variables without confusion.

Equation Simplification

Equation simplification is an essential part of solving equations accurately. It involves performing algebraic manipulations to make the equations more manageable.

In our exercise, the first step is to simplify the first equation by isolating terms involving y. This helps us to easily substitute values later.

• Start with the equation: 5x + 2y = 4y + 9.
• Move all terms involving y to one side: 5x + 2y - 2y = 4y + 9 - 2y.
• Simplify it to: 5x = 2y + 9.

After simplifying the equation, the next step is to substitute the expression for y provided by the second equation (y = x - 3). This substitution simplifies the first equation, enabling us to combine like terms and isolate x:

• Substitute y in the first equation: 5x = 2(x - 3) + 9.
• Expand and simplify: 5x = 2x - 6 + 9, which becomes: 5x = 2x + 3.
• Isolate x by subtracting 2x from both sides: 3x = 3.
• Finally, solve for x by dividing both sides by 3: x = 1.

Simplification allows us to handle complex equations step by step, making it easier to find solutions.

Solution Verification

Verifying the solution is a crucial step to ensure the values obtained are correct and satisfy the original equations. In our exercise, after finding x and y, we substitute these values back into the original system to check for consistency.

• First, we found x = 1 and y = -2.
• Substitute x = 1 and y = -2 in the original equations to verify: 5(1) + 2(-2) = 4(-2) + 9.
• Simplify to see if both sides of the equation equal: 5 - 4 = -8 + 9, which simplifies to 1 = 1.

Both sides of the equation must be equal for the values to be correct. This confirmation step reassures us that our calculations were accurate. If the left-hand side and right-hand side of the equations equal each other after substitution, it means the values of x and y we obtained are indeed the correct solutions to the system.

Always remember to perform this verification to avoid any mistakes in the solution.