Solve for x 7x+(3x)/2=-17

The problem provided is a linear equation with a single variable, x. The equation includes a multiplication of x by 7, an addition of three halves of x, and equates this sum to a negative number, -17. The objective is to manipulate the equation using algebraic operations to isolate x on one side of the equation, thus finding the value of x that makes the equation true. This involves combining like terms and undertaking operations such as division or multiplication to solve for the unknown variable x.

$7 x + \frac{3 x}{2} = - 17$

Answer

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Solution:

Step:1

Transform $7 x + \frac{3 x}{2}$ into a single fraction.

Step:1.1

Convert $7 x$ to a fraction by multiplying by $\frac{2}{2}$ : $7 x \cdot \frac{2}{2} + \frac{3 x}{2} = - 17$

Step:1.2

Simplify the expression.

Step:1.2.1

Merge $7 x$ with $\frac{2}{2}$ : $\frac{7 x \cdot 2}{2} + \frac{3 x}{2} = - 17$

Step:1.2.2

Combine like terms over the shared denominator: $\frac{7 x \cdot 2 + 3 x}{2} = - 17$

Step:1.3

Refine the numerator.

Step:1.3.1

Extract $x$ from $7 x \cdot 2 + 3 x$ .

Step:1.3.1.1

Extract $x$ from $7 x \cdot 2$ : $\frac{x (7 \cdot 2) + 3 x}{2} = - 17$

Step:1.3.1.2

Extract $x$ from $3 x$ : $\frac{x (7 \cdot 2) + x \cdot 3}{2} = - 17$

Step:1.3.1.3

Take $x$ out of $x (7 \cdot 2) + x \cdot 3$ : $\frac{x (7 \cdot 2 + 3)}{2} = - 17$

Step:1.3.2

Calculate $7$ times $2$ : $\frac{x (14 + 3)}{2} = - 17$

Step:1.3.3

Add $14$ and $3$ : $\frac{x \cdot 17}{2} = - 17$

Step:1.4

Reposition $17$ to precede $x$ : $\frac{17 x}{2} = - 17$

Step:2

Multiply the equation by $\frac{2}{17}$ to isolate $x$ : $\frac{2}{17} \cdot \frac{17 x}{2} = \frac{2}{17} \cdot - 17$

Step:3

Simplify both sides of the equation.

Step:3.1

Clarify the left side.

Step:3.1.1

Simplify $\frac{2}{17} \cdot \frac{17 x}{2}$ .

Step:3.1.1.1

Eliminate the common factor of $2$ : $\frac{2}{17} \cdot \frac{17 x}{2} = \frac{2}{17} \cdot - 17$

Step:3.1.1.2

Eliminate the common factor of $17$ : $\frac{1}{17} (17 x) = \frac{2}{17} \cdot - 17$

Step:3.2

Streamline the right side.

Step:3.2.1

Simplify $\frac{2}{17} \cdot - 17$ .

Step:3.2.1.1

Remove the common factor of $17$ : $x = \frac{2}{17} \cdot (17 \cdot - 1)$

Step:3.2.1.2

Multiply $2$ by $- 1$ : $x = - 2$

Knowledge Notes:

The problem involves solving a linear equation with a single variable, $x$ . The equation contains both whole numbers and fractions, which requires finding a common denominator to combine like terms.

Key knowledge points include:

Fractions: Understanding how to manipulate fractions, including finding a common denominator and simplifying.
Combining Like Terms: The process of merging terms that have the same variable to simplify an expression.
Isolating the Variable: The goal in solving an equation is to isolate the variable on one side of the equation. This often involves performing the same operation on both sides of the equation.
Simplification: Reducing an expression to its simplest form by performing arithmetic operations and canceling common factors.
Multiplication and Division Principles: When solving equations, multiplying or dividing both sides by the same non-zero number does not change the equality.

In this problem, we use these principles to first combine like terms by finding a common denominator, then factor out the variable $x$ , and finally isolate $x$ by multiplying both sides by the reciprocal of the coefficient of $x$ . The solution is found by simplifying the resulting expression.