Simplify the expression 1.54x – 4(2.5x + 3) + 8.2x.
The Correct Answer and Explanation is :
To simplify the expression (1.54x – 4(2.5x + 3) + 8.2x), we will follow the steps of distribution, combining like terms, and final simplification.
Step 1: Distribute the ( -4 )
The first part of the expression involves distributing the (-4) across the terms inside the parentheses.
[
-4(2.5x + 3) = -4 \cdot 2.5x – 4 \cdot 3 = -10x – 12
]
Step 2: Substitute Back into the Expression
Now, we replace the distributed term back into the original expression:
[
1.54x – 10x – 12 + 8.2x
]
Step 3: Combine Like Terms
Next, we will combine the (x) terms:
- The (x) terms are (1.54x), (-10x), and (8.2x).
Adding them together:
[
1.54x – 10x + 8.2x = (1.54 – 10 + 8.2)x
]
Calculating the coefficients:
[
1.54 – 10 + 8.2 = 1.54 + 8.2 – 10 = 9.74 – 10 = -0.26
]
Thus, the combined (x) term simplifies to:
[
-0.26x
]
Now, we include the constant term:
[
-12
]
Step 4: Final Expression
Putting everything together, the simplified expression is:
[
-0.26x – 12
]
Explanation
In this simplification process, we first applied the distributive property to eliminate the parentheses, which is crucial for accurately simplifying expressions that include multiplication and addition. After distributing, we gathered all like terms, which are terms that contain the same variable (in this case, (x)).
Combining coefficients allows us to consolidate the expression into a simpler form, making it easier to understand and work with. Finally, including constant terms helps achieve a fully simplified linear expression. The final result, ( -0.26x – 12 ), represents a linear equation in slope-intercept form, which can be useful for graphing or further mathematical manipulations.