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

Solution:

Step:1

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

Step:1.1

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

Step:1.2

Simplify the expression.

Step:1.2.1

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

Step:1.2.2

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

Step:1.3

Refine the numerator.

Step:1.3.1

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

Step:1.3.1.1

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

Step:1.3.1.2

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

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 $x$: $\frac{17x}{2} = -17$

Step:2

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

Step:3

Simplify both sides of the equation.

Step:3.1

Clarify the left side.

Step:3.1.1

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

Step:3.1.1.1

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

Step:3.1.1.2

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

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.