Question

For the following equation, complete the related subtraction equation used to solve the equation. \(\begin{aligned} x+10 &=-9 \\ x &=\end{aligned}\) Solve and check. $$-36+x=-53$$

Short answer

Answer: The solutions are \(x=-19\) and \(x=-17\).

Step-by-step solution

Solve the first equation

To solve \(x+10=-9\), we need to isolate \(x\). To do that, we will subtract \(10\) from both sides of the equation: \(\begin{aligned} x+10-10&=-9-10 \\ x&=-19 \end{aligned}\) The related subtraction equation used to solve this equation is: \(x+10-10= -9-10\) Now, we will solve the second equation.

Solve the second equation

To solve \(-36+x=-53\), we need to isolate \(x\). To do that, we will add \(36\) to both sides of the equation: \(\begin{aligned} -36+x+36&=-53+36 \\ x&=-17 \end{aligned}\) The related subtraction equation used to solve this equation is: \(-36+x+36= -53+36\)

Check our solutions

To check our solutions, we plug our values of \(x\) into the original equations. For the first equation: \(\begin{aligned} x+10&=-9 \\ (-19)+10&=-9 \\ -9&=-9 \end{aligned}\) This confirms our solution for the first equation. For the second equation: \(\begin{aligned} -36+x&=-53 \\ -36+(-17)&=-53 \\ -53&=-53 \end{aligned}\) This confirms our solution for the second equation. The solutions to both equations are \(x=-19\) and \(x=-17\).

Key concepts

Algebraic Manipulation

Algebraic manipulation is the process of rearranging an equation to isolate the variable you are solving for. This usually involves performing operations such as addition, subtraction, multiplication, or division. The key is to perform the same operation on both sides of the equation to keep it balanced. Imagine you have the equation \( x + 10 = -9 \). To isolate \(x\), you subtract \(10\) from both sides. This process looks like this:

• Start with the original equation: \( x + 10 = -9 \)
• Subtract \(10\) from both sides: \( x + 10 - 10 = -9 - 10 \)
• This simplifies to: \( x = -19 \)

The main goal is to get \(x\) by itself on one side of the equation. Algebraic manipulation allows you to see the relationship between \(x\) and the constant value on the other side.

Checking Solutions

Once you have found a solution to an equation, it is essential to verify that it is correct. This process is known as checking solutions. By plugging the solution back into the original equation, you can ensure that both sides are equal. Take the equation we solved earlier \( x + 10 = -9 \) with the solution \( x = -19 \). To check this:

• Substitute \(-19\) for \(x\) in the equation: \((-19) + 10 = -9 \)
• Simplify the equation: \(-9 = -9\)
• The equation holds true, confirming our solution is correct.

This simple step provides confidence in your work, ensuring that no mistakes were made in the solving process.

Inverse Operations

Inverse operations are operations that undo each other. They are fundamental tools for solving equations because they help isolate the variable. The basic inverse operations are:

• Addition and subtraction
• Multiplication and division

For example, in the equation \(-36 + x = -53\), we want to isolate \(x\). Notice that \(-36\) is being added to \(x\). The inverse operation is addition, so we add \(36\) to both sides:

• Start with the equation: \(-36 + x = -53\)
• Add \(36\) to both sides: \(-36 + x + 36 = -53 + 36\)
• This simplifies to: \(x = -17\)

The use of inverse operations lets us systematically and effectively solve equations by balancing the changes made on both sides of an equation.