Solve for y 17-y=6(y-3)

Your problem is a linear equation that requires you to solve for the variable y. The equation is set up with a term containing y on both sides of the equal sign, which will necessitate moving terms that contain y to one side and constants to the other to isolate y. By applying algebraic manipulations such as distribution, subtraction, and division, you aim to find the value of y that satisfies the equation.

$17 - y = 6 (y - 3)$

Answer

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Solution:

Step 1

Expand the expression $6 (y - 3)$ .

Step 1.1

Express the equation as $17 - y = 0 + 0 + 6 (y - 3)$ .

Step 1.2

Incorporate zero additions into the equation: $17 - y = 6 (y - 3)$ .

Step 1.3

Use the distributive property to expand: $17 - y = 6 y - 6 \cdot 3$ .

Step 1.4

Carry out the multiplication: $17 - y = 6 y - 18$ .

Step 2

Isolate terms with $y$ on one side of the equation.

Step 2.1

Subtract $6 y$ from both sides: $17 - y - 6 y = - 18$ .

Step 2.2

Combine like terms: $17 - 7 y = - 18$ .

Step 3

Move constant terms to the opposite side.

Step 3.1

Subtract $17$ from both sides: $- 7 y = - 18 - 17$ .

Step 3.2

Combine the constants: $- 7 y = - 35$ .

Step 4

Solve for $y$ by dividing both sides by $- 7$ .

Step 4.1

Divide the equation by $- 7$ : $\frac{- 7 y}{- 7} = \frac{- 35}{- 7}$ .

Step 4.2

Simplify the left side by canceling out $- 7$ .

Step 4.2.1

Eliminate the common factor: $\frac{- 7 y}{- 7} = \frac{- 35}{- 7}$ .

Step 4.2.1.1

Simplify to get $y$ : $y = \frac{- 35}{- 7}$ .

Step 4.3

Simplify the right side by dividing: $y = 5$ .

Knowledge Notes:

To solve the equation $17 - y = 6 (y - 3)$ , we follow these steps:

Distributive Property: This property states that $a (b + c) = a b + a c$ . We use it to expand $6 (y - 3)$ into $6 y - 18$ .
Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power. For example, $- y - 6 y$ becomes $- 7 y$ .
Isolating the Variable: We rearrange the equation to have all terms with the variable on one side and constants on the other. This often involves adding or subtracting terms from both sides of the equation.
Solving the Equation: Once we have isolated the variable, we can solve for it by performing operations that will leave the variable alone on one side of the equation. In this case, we divide both sides by $- 7$ to solve for $y$ .
Checking the Solution: It's always good practice to check the solution by plugging it back into the original equation to ensure it satisfies the equation.

In this problem, we applied these principles to solve for $y$ and found that $y = 5$ .