Problem 62 - Algebra 2

Question

Solve each system of equations. $$ \begin{array}{l}{3 x-2 y=-2} \\ {4 x+7 y=65}\end{array} $$

Step-by-Step Solution

Answer

The solution for the system is \((x, y) = (4, 7)\).

Step 1: Choose a Method

We will use the method of substitution to solve the given system of equations.

Step 2: Solve for One Variable

First, we will solve the first equation for \( x \). The equation is \( 3x - 2y = -2 \). We can express \( x \) in terms of \( y \) by adding \( 2y \) to both sides and then dividing by 3. \[ x = \frac{2y - 2}{3} \]

Step 3: Substitute Expression into Second Equation

Next, substitute the expression for \( x \) that we found into the second equation \( 4x + 7y = 65 \). This gives us: \[ 4\left(\frac{2y - 2}{3}\right) + 7y = 65 \]

Step 4: Simplify and Solve for \( y \)

Multiply through by 4: \[ \frac{8y - 8}{3} + 7y = 65 \] Clear the fraction by multiplying each term by 3: \[ 8y - 8 + 21y = 195 \] Combine like terms: \[ 29y - 8 = 195 \] Add 8 to both sides: \[ 29y = 203 \] Finally, divide by 29: \[ y = 7 \]

Step 5: Substitute \( y \) back to Find \( x \)

Now substitute \( y = 7 \) back into the expression for \( x \): \[ x = \frac{2(7) - 2}{3} \] Calculate: \[ x = \frac{14 - 2}{3} = \frac{12}{3} = 4 \]

Step 6: Verify the Solution

Plug \( x = 4 \) and \( y = 7 \) back into the original equations to verify: For the first equation: \[ 3(4) - 2(7) = 12 - 14 = -2 \] For the second equation: \[ 4(4) + 7(7) = 16 + 49 = 65 \] Both equations are satisfied, so the solution is correct.

Key Concepts

Substitution Method

The substitution method is a common technique for solving systems of equations. It involves expressing one variable in terms of another and substituting that expression into the other equation.
This allows us to work with only one equation and one unknown, simplifying the process.

• Start with one of the given equations.
• Solve the equation for one variable to express it in terms of the other.
• Substitute this expression into the other equation.

This method is especially useful when one of the equations can be easily rearranged to isolate one variable.
As seen in our example, we initially solved for "\( x \)" in terms of "\( y \)." This provided a clear path to then substitute back into the second equation.

Solving Linear Equations

Solving linear equations involves finding the values of the variables that make the equation true. When using the substitution method, after substitution, we simplify the resulting equation.
Here are the steps generally involved:

• Combine like terms when necessary.
• Add or subtract terms to isolate the variable expression on one side.
• Multiply or divide all terms by the coefficient of that variable to solve for the variable.

In our case, once the expression for "\( x \)" was substituted into the second equation, we simplified and solved for "\( y \)" by clearing the fraction and combining like terms.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging equations or expressions to simplify them or isolate a variable.
This might involve using basic arithmetic operations like addition, subtraction, multiplication, or division. Simplifying equations is crucial to make them easier to solve.

• Start by clearing fractions or decimals if present. Multiplying all terms by a common denominator can help.
• Combine like terms to simplify expressions.
• Use inverse operations to isolate variables.

In our system, we used algebraic manipulation to derive an expression for "\( x \)," then simplified the equation for "\( y \)." This simplification was crucial for easily finding the solution.

Verification of Solutions

Verification of solutions involves plugging the solutions back into the original equations to ensure they hold true. This confirms the correctness of the solution.
This is a critical step in solving systems of equations to prevent errors.

• Substitute the found values back into each of the original equations.
• Check that both sides of each equation are equal.
• If all equations are satisfied, the solution is correct.

In our example, we verified the solution by plugging "\( x = 4 \)" and "\( y = 7 \)" back into both original equations. Both verifications confirmed the accuracy of the solutions, solidifying our methodology.