Solve for y 18-y=4(y-3)
In this problem, you are asked to determine the value of the variable "y" by solving the given linear equation. The equation provided is "18 - y = 4(y - 3)". The solution process will involve applying algebraic manipulations such as distributing the multiplication over subtraction within the parentheses, moving terms involving the variable "y" to one side of the equation, and numerical terms to the other side, and then solving for "y" by isolating it.
$18 - y = 4 \left(\right. y - 3 \left.\right)$
Answer
Solution:
Step 1: Expand the expression on the right side of the equation.
Step 1.1: Start with the original equation. $18 - y = 4(y - 3)$ Step 1.2: Recognize that adding zero does not change the value. $18 - y = 0 + 4(y - 3)$ Step 1.3: Use the distributive property to expand the multiplication. $18 - y = 4y - 4 \cdot 3$ Step 1.4: Perform the multiplication. $18 - y = 4y - 12$
Step 2: Isolate the variable terms on one side.
Step 2.1: Subtract $4y$ from both sides to move variable terms to one side. $18 - y - 4y = -12$ Step 2.2: Combine like terms involving $y$. $18 - 5y = -12$
Step 3: Isolate the constant terms on the opposite side.
Step 3.1: Subtract $18$ from both sides to move constants to the other side. $-5y = -12 - 18$ Step 3.2: Combine the constants. $-5y = -30$
Step 4: Solve for the variable.
Step 4.1: Divide both sides by $-5$ to isolate $y$. $\frac{-5y}{-5} = \frac{-30}{-5}$ Step 4.2: Simplify the equation.
Step 4.2.1: Reduce the fraction on the left side. $\frac{\cancel{-5}y}{\cancel{-5}} = \frac{-30}{-5}$ Step 4.2.1.1: Simplify to $y$. $y = \frac{-30}{-5}$ Step 4.3: Simplify the right side by dividing. $y = 6$
Knowledge Notes:
The problem-solving process involves algebraic manipulation to solve for the variable $y$ in the equation $18 - y = 4(y - 3)$. The steps taken are common algebraic techniques used to isolate the variable and solve the equation.
Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to expand expressions that involve multiplication over addition or subtraction.
Combining Like Terms: This refers to the process of adding or subtracting terms that have the same variable raised to the same power. In this case, $-y$ and $-4y$ are combined to get $-5y$.
Isolating the Variable: The goal is to get the variable by itself on one side of the equation. This often involves moving terms from one side of the equation to the other by performing the same operation on both sides.
Solving the Equation: Once the variable is isolated, you can solve for its value by performing any necessary arithmetic operations. In this case, dividing both sides by $-5$ gives the value of $y$.
Checking the Solution: Although not explicitly stated in the steps, it is good practice to check the solution by substituting the value of $y$ back into the original equation to ensure that both sides are equal.
