Problem 134 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 3 x+2 y=-3 \\ -x-2 y=-19 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = -11\) and \(y = 15\).

Step 1 - Solve one equation for one variable

First, solve the second equation for one variable. Let's solve for x: \(-x - 2y = -19\)Rearrange the equation: \(x = -2y + 19\)

Step 2 - Substitute expression into the other equation

Substitute \(x = -2y + 19\) into the first equation: \(3(-2y + 19) + 2y = -3\)

Step 3 - Simplify and solve for y

Distribute and simplify the equation: \(-6y + 57 + 2y = -3\)Combine like terms: \(-4y + 57 = -3\)Subtract 57 from both sides: \(-4y = -60\)Divide both sides by -4: \(y = 15\)

Step 4 - Solve for x using y = 15

Use \(y = 15\) in the expression \(x = -2y + 19\): \(x = -2(15) + 19\)Simplify: \(x = -30 + 19\)\(x = -11\)

Step 5 - Verify the solution

Substitute \(x = -11\) and \(y = 15\) into both original equations to check if they satisfy both: 1) \(3(-11) + 2(15) = -3\)\(-33 + 30 = -3\), which is true. 2) \(-(-11) - 2(15) = -19\)\(11 - 30 = -19\), which is also true.

Key Concepts

substitution method

The substitution method is a straightforward procedure used to solve systems of equations. In this method, one of the equations is solved for one variable in terms of the other variable. Then, this expression is substituted into the other equation. This allows you to solve for one variable at a time, simplifying the system step by step.

The main steps include:

• Solve one equation for one variable.
• Substitute this expression into the other equation.
• Simplify and solve for the other variable.
• Substitute back to find the first variable.

In the given exercise, we solved the system by first isolating x in the second equation, ewline \(-x - 2y = -19\) to get \( x = -2y + 19 \). Next, we substituted this expression for x in the first equation to find y.

solving linear equations

Solving linear equations involves finding the value of the variable that makes the equation true. In the context of the substitution method, after substitution, you are left with a linear equation with one variable.

In our example:

• Starting with \( 3(-2y + 19) + 2y = -3 \)
• We distribute to get \( -6y + 57 + 2y = -3 \)
• Combine like terms to obtain \( -4y + 57 = -3 \)
• Subtract 57 from both sides giving \( -4y = -60 \)
• Finally, divide both sides by -4 to get \( y = 15 \)

Once you’ve solved for y, you substitute it back into the expression for x to solve for the second variable.

verification of solutions

Verification of solutions is an essential step to ensure the results are correct. After solving for both variables, substitute them back into the original equations and check if both equations hold true.

In our exercise, we found that \( x = -11 \) and \( y = 15 \). Let's verify:

• First equation: \( 3(-11) + 2(15) = -3 \)
• Simplifies to \( -33 + 30 = -3 \), which is true.
• Second equation: \( -(-11) - 2(15) = -19 \)
• Simplifies to \( 11 - 30 = -19 \), which is also true.

If both equations are satisfied, it confirms that the solution is correct. Verification ensures accuracy and builds confidence.