Question

Solve the equation. $$55 x-3(9 x+12)=-64$$

Short answer

The solution to the equation is \(x = -1\).

Step-by-step solution

Use distributive property

Apply the distributive property to \(3(9x+12)\) on the left side of the equation. This means multiplying each term within the parenthesis by 3: \(55x - 3*9x - 3*12 = -64\). This simplifies to \(55x - 27x - 36 = -64\).

Combine like terms

On the left side of the equation, combine the terms involving \(x\): \(28x - 36 = -64\)

Solve for \(x\)

Finally, isolate \(x\). Add 36 to both sides of the equation, giving us \(28x = -64 + 36\), which further simplifies to \(28x = -28\). Divide both sides by 28 to isolate \(x\), which gives us \(x = -1\)

Key concepts

Distributive Property

The distributive property is a fundamental algebraic principle that allows us to simplify expressions and solve equations efficiently. It states that a number outside the parenthesis, multiplied by the sum or difference within the parenthesis, can be 'distributed' across each term inside.

For instance, in our exercise, we are given the term \(3(9x+12)\). By applying the distributive property, we multiply 3 with each term inside the parentheses: \(3 \times 9x\) and \(3 \times 12\), which simplifies to \(27x + 36\), but remember we have a minus before the parenthesis, so it becomes \( -27x - 36\). This step transforms the equation into a more manageable form without parentheses, making it easier to see the 'like terms' that we will move on to combine in the next step.

Combine Like Terms

Combining like terms is the process of simplifying an algebraic expression or equation by adding or subtracting terms that have the same variable raised to the same power. Like terms can also be constants.

After the distributive step in the exercise, our equation looks like \(55x - 27x - 36=-64\). Here, the terms \(55x\) and \( -27x\) are like terms since they both are coefficients of the same variable \(x\). We combine them by simply adding or subtracting their coefficients to get \(28x - 36 = -64\). This simplification makes the equation more straightforward and prepares us for the final step of isolating the variable.

Isolation of Variable

Isolating the variable is the essential last step in solving linear equations. It refers to the process of manipulating the equation so that the variable we are solving for (in this case, \(x\)) is alone on one side of the equation.

Continuing from the previous steps, we have the simplified equation \(28x - 36 = -64\). To isolate \(x\), we need to get rid of the -36 by adding 36 to both sides of the equation, resulting in \(28x = -28\). At this point, \(x\) is almost isolated, but it's currently multiplied by 28. To complete the isolation, we divide both sides of the equation by 28, giving us the solution \(x = -1\). This technique is crucial because it allows us to find the specific value of the variable that satisfies the equation.