Solve for y x=2-y

Solution:

Step 1:

Express the equation in the form $x=2-y$.

Step 2:

Isolate the term containing $y$ by subtracting $2$ from both sides, yielding $x-2=-y$.

Step 3:

Eliminate the negative sign in front of $y$ by dividing the entire equation by $-1$.

Step 3.1:

Apply the division to each term: $\frac{x}{-1}-\frac{2}{-1}=\frac{-y}{-1}$.

Step 3.2:

Resolve the division on the left-hand side.

Step 3.2.1:

Recognize that dividing a negative by a negative gives a positive: $y=\frac{x}{-1}-\frac{2}{-1}$.

Step 3.2.2:

Simplify the division of $y$ by $1$: $y=\frac{x}{-1}-\frac{2}{-1}$.

Step 3.3:

Simplify the right-hand side.

Step 3.3.1:

Handle each term individually.

Step 3.3.1.1:

Convert the division of $x$ by $-1$ to multiplication: $y=-x-\frac{2}{-1}$.

Step 3.3.1.2:

Express $-1\cdot x$ as $-x$: $y=-x-\frac{2}{-1}$.

Step 3.3.1.3:

Simplify the division of $-2$ by $-1$: $y=-x+2$.

Thus, the solution to the equation $x=2-y$ is $y=-x+2$.

Knowledge Notes:

To solve for a variable in an equation, you need to isolate the variable on one side of the equation. This often involves performing the same mathematical operations on both sides of the equation to maintain equality. Here are the relevant knowledge points for this problem:

1. Rearranging Equations: Equations can be rewritten in different forms as long as the equality is preserved. For example, $x=2-y$ can be rewritten as $2-y=x$.

2. Solving Linear Equations: To solve for a variable, you need to get the variable by itself on one side of the equation. This may involve adding, subtracting, multiplying, or dividing both sides of the equation by the same number.

3. Division and Multiplication by Negative Numbers: Dividing or multiplying both sides of an equation by a negative number will change the sign of each term. For instance, dividing $-y$ by $-1$ yields $y$.

4. Simplification: After performing operations, it's important to simplify the equation. This can include combining like terms, reducing fractions, or simplifying expressions.

5. Inverse Operations: To undo an operation, you use its inverse. For example, to undo subtraction, you add the same number to both sides of the equation.

By understanding and applying these principles, you can solve linear equations and isolate the desired variable.