Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. Equations Having Symbols of Grouping. $$5-3(x-2)=4(3 x-1)$$

Short answer

x = 1

Step-by-step solution

Distribute the Negative Across the Parentheses

Multiply -3 across the parentheses: -3 times (x) gives -3x, and -3 times (-2) gives +6. The equation then becomes: 5 - 3x + 6 = 4(3x - 1).

Simplify Both Sides

Combine like terms on the left side of the equation:5 + 6 becomes 11, so the equation simplifies to:11 - 3x = 4(3x - 1).

Distribute the 4 on the Right Side

Multiply 4 across the parentheses:4 times 3x gives 12x, and 4 times -1 gives -4.This transforms the equation into:11 - 3x = 12x - 4.

Move the Variable Terms to One Side

Add 3x to both sides to get:11 = 15x - 4.

Isolate the Variable

Add 4 to both sides so that all constants are on one side and the variable is on the other:11 + 4 = 15x, so15 = 15x.

Divide by the Coefficient of the Variable

Divide both sides by 15 to solve for x:15 / 15 = 15x / 15,thus x = 1.

Key concepts

Symbols of Grouping

When solving linear equations, symbols of grouping such as parentheses ( ), brackets [ ], and braces { }, play a significant role. They indicate which operations should be performed first according to the order of operations—PEMDAS/BODMAS.

For example, in the equation 5 - 3(x - 2) = 4(3x - 1), the parentheses tell us to perform the calculations inside them before carrying out other operations. This is why in Step 1 of the solution, we multiply -3 across the x and -2 inside the parentheses. This step is crucial because ignoring it or misapplying the distributive property could lead to an incorrect solution.

Remember, when dealing with symbols of grouping:

• Perform operations inside the grouping symbols first.
• Use the distributive property to eliminate the parentheses.
• Be mindful of sign changes when multiplying or dividing.

Equation Simplification

The next step in solving linear equations is equation simplification. Simplification makes the equation easier to solve by combining like terms and reducing expressions to their simplest form.

In our example, after distributing, we combined the constant terms (5 and 6) on the left side to simplify the equation to 11 - 3x = 4(3x - 1). We then further simplified by distributing the 4 on the right side of the equation. Simplification is a critical step for two main reasons:

• It reduces the potential for errors in subsequent steps.
• It makes the structure of the equation clearer, setting the stage for isolating the variable.

Key tips for success in equation simplification include:

• Always combine like terms where possible.
• Ensure to carry over each term's sign during simplification, like converting -3 times -2 into +6 and not 6.
• Double-check your work to make sure all terms have been accounted for correctly.

Variable Isolation

Finally, to solve an equation, we need to isolate the variable; that means getting the variable on one side of the equality and the constants on the other. As shown from Step 4 to Step 6 in our example, we moved all terms with x to one side and constants to the opposite side.

Here is the process:

• Add or subtract terms to get all variables on one side and constants on the other side. In the given problem, we added 3x to both sides which eliminated -3x from the left.
• Further simplify if necessary, as done by adding 4 to both sides in our example.
• Finally, divide by the variable's coefficient to solve for the variable, leaving us with x = 1.

Remember, variable isolation is the step where the solution becomes clear. It involves reverse operations—if the variable is multiplied by a number, we divide by that number; if it's divided by a number, we multiply; and similarly, add or subtract to undo any addition or subtraction of numbers from the variable. This systematic approach leads to the correct value of the variable that satisfies the original equation.