Answered

Simplify the following expressions or solve the equations:


  1. If $50x = 5$ and $x = 42 \div 7 = 6$, so $x = 6$.


ASimplify:

$x + 2x + 3x$


BSimplify:

$3a + 3 + 2a + 1$

Verified Answer :

  • For part (a), the equation \(50x=50\) yields \(x=1\), while the calculation \(42 \div 7\) yields \(x=6\).
  • For part (b), combining like terms in \(x + 2x + 3x\) results in \(6x\).
  • For part (c), grouping and summing like terms in \(3a + 3 + 2a + 1\) results in \(5a + 4\).
  • Final results: \(\boxed{x=1, x=6; 6x; 5a+4}\).

Explanation

  1. Analyzing and Solving the Algebraic Expressions

The problem provides three parts involving basic algebraic operations and simplifications. We will analyze each part individually.

ASolving for \(x\):

We are given the equation \(50x = 50\). To solve for \(x\), divide both sides by \(50\):
\[x = \frac{50}{50} = 1\]
Wait, the provided text says "x = 42 \div 7 = 6, so x = 6". This seems to be describing two different problems or a sequence of operations. Looking closely:

  • For \(50x = 50\), the solution is \(x = 1\).
  • For \(x = 42 \div 7\), the solution is \(x = 6\).

The provided text presents these as separate operations or examples. We will treat them as independent calculations.

BSimplifying the expression \(x + 2x + 3x\):

We combine like terms. The coefficients of \(x\) are \(1, 2,\) and \(3\).
\[x + 2x + 3x = (1 + 2 + 3)x = 6x\]
This matches the provided result.

CSimplifying the expression \(3a + 3 + 2a + 1\):

We group the variable terms and the constant terms separately.

  • Variable terms: \(3a + 2a = 5a\)
  • Constant terms: \(3 + 1 = 4\)

Combining these, we get \(5a + 4\). This matches the provided result.

Examples

Algebraic manipulation is fundamental in everyday problem solving. For example, if you are calculating the total cost of a shopping trip where you buy 3 notebooks at 'a' dollars each, 2 pens at 'a' dollars each, plus a shipping fee of \$3 and a tax of \$1, the total cost would be \(3a + 3 + 2a + 1\). By simplifying this to \(5a + 4\), you can quickly determine the total price for any given cost 'a' of the notebooks and pens.

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