Solving Algebraic Equations and Simplifying Fractions

In this article, we will work through the process of solving an algebraic equation and simplifying fractions. Specifically, we will solve the equation 2(3x - 7) - 4(3x 2) 6(5x - 9) 3. Let's break down the steps involved in solving this equation and simplifying the result.

Step-by-Step Solution

First, we will simplify the left-hand side and the right-hand side of the equation using the distributive property of multiplication. 2(3x - 7) - 4(3x 2) 6(5x - 9) 3 By applying the distributive property, we get: 6x - 14 - 12x - 8 3 - 54 3 Next, we combine like terms on both sides of the equation. 6x - 12x - 14 - 8 3 - 51 Simplifying the left-hand side, we have: -6x - 22 3 - 51 To isolate the variable ( x ), we move all the terms with ( x ) to one side and the constants to the other side. We start by adding ( 6x ) to both sides. -22 36x - 51 Next, we add 51 to both sides to isolate the term with ( x ). 29 36x Finally, we divide both sides by 36 to solve for ( x ). x frac{29}{36}

Verification

To verify our solution, let's substitute ( x frac{29}{36} ) back into the original equation and check if both sides are equal. 2(3(frac{29}{36}) - 7) - 4(3(frac{29}{36}) 2) 6(5(frac{29}{36}) - 9) 3 Simplifying the expression on the left-hand side: 2(3(frac{29}{36}) - 7) - 4(3(frac{29}{36}) 2) 2( frac{87}{36} - 7) - 4( frac{87}{36} 2) 2( frac{87}{36} - frac{252}{36}) - 4( frac{87}{36} frac{72}{36}) 2( -frac{165}{36}) - 4( frac{159}{36}) 2( -frac{165}{36}) - 4( frac{159}{36}) -frac{330}{36} - frac{636}{36} -frac{966}{36} Simplifying the expression on the right-hand side: 6(5(frac{29}{36}) - 9) 3 6( frac{145}{36} - 9) 3 6( frac{145}{36} - frac{324}{36}) 3 6( -frac{179}{36}) 3 6( -frac{179}{36}) 3 -frac{1074}{36} 3 -frac{1074}{36} frac{108}{36} -frac{966}{36} Both sides are equal, confirming that our solution ( x frac{29}{36} ) is correct.

Fraction Simplification

In fraction simplification, let's convert the solution ( x frac{29}{36} ) to a mixed number and a decimal.

As a Mixed Number

Divide 29 by 36:

29 ÷ 36 0 with a remainder of 29

Therefore, ( x frac{29}{36} ) as a mixed number is ( 0 frac{29}{36} ) or simply ( 0 frac{29}{36} ).

As a Decimal

( frac{29}{36} approx 0.805556 ) (rounded to 6 decimal places)

Thus, the decimal representation of ( x ) is approximately 0.805556 or approximately ( 0.81 ) when rounded to two decimal places.