Question

If $3(x-6)+8-(2-4x)=7-4(x+2),$ what is the value of $x?$

Short answer

The value of x is 1.

Step-by-step solution

- Distribute and Simplify

First, distribute the constants through the parentheses. For the left side: 1) Expand: - Distribute 3: 3(x - 6) + 8 - (2 - 4x) = 3x - 18 + 8 - 2 + 4x. For the right side: 2) Distribute -4: 7 - 4(x + 2) = 7 - 4x - 8.

- Combine Like Terms

Combine like terms on both sides of the equation: For the left side: 3x - 18 + 8 - 2 + 4x = 7x - 12. For the right side: 7 - 4x - 8 = -4x - 1.

- Move All Terms Involving x to One Side

Add 4x to both sides of the equation to combine the x terms: 7x - 12 + 4x = -4x - 1 + 4x 11x - 12 = -1.

- Isolate the Variable

Solve for x by isolating the variable. First, add 12 to both sides: 11x - 12 + 12 = -1 + 12 . Thus: 11x = 11. Finally, divide both sides by 11: x = 1.

Key concepts

Distributive Property

When solving linear equations, one key concept is the distributive property. This property allows you to distribute a single term across terms inside parentheses. For instance, in the equation $3(x-6)+8-(2-4x)=7-4(x+2)$, we apply the distributive property to simplify:

• First, distribute 3 through $x-6$: $$3(x-6)=3x-18$$.
• In the term $-(2-4x)$, distribute the negative sign, which changes signs of the terms inside the parentheses:$$-(2-4x)=-2+4x$$.
• On the right side, distribute -4 through $x+2$:$$-4(x+2)=-4x-8$$.

Applying the distributive property helps to eliminate parentheses, turning the original equation into a line of terms: $$3x-18+8-2+4x=7-4x-8$$.

Combining Like Terms

After using the distributive property, the next step in solving our linear equation is to combine like terms. Like terms are terms that contain the same variables raised to the same power.

• On the left side of our equation, combine the terms with $x$:$$3x+4x=7x$$.
• Next, combine the constant terms:$$-18+8-2=-12$$.
• The left side simplifies to $$7x-12$$.
• On the right side, combine the constant terms:$$7-8=-1$$.

Now, the simplified equation looks like this:$$7x-12=-4x-1$$. Combining like terms helps consolidate the equation, making it easier to solve.

Isolating Variable x

To solve for $x$, we need to get all the $x$ terms on one side of the equation. Follow these steps:

• Add $4x$ to both sides of the equation to move the $x$ terms to one side:$$7x-12+4x=-4x-1+4x$$.
• Simplify to get:$$11x-12=-1$$.

Isolating the $x$ variable helps create a straightforward equation to solve for the variable.

Algebraic Manipulation

The final step involves basic algebraic manipulation to isolate $x$.

First, add 12 to both sides to move the constant term:

• $$11x-12+12=-1+12$$.
• This simplifies to:$$11x=11$$.

Next, divide both sides by 11:

• $$\frac{11x}{11}=\frac{11}{11}$$
• Which simplifies to $$x=1$$.

Combining all these steps, you can solve the equation easily by isolating the variable and applying basic algebraic operations.