Problem 38 - Algebra 1

Question

Solve the equation $$ -3 y+14=-5 y $$

Step-by-Step Solution

Answer

The solution to the equation is \(y = -7\)

Step 1: Collect like terms

In any equation, the first process is to collect like terms together. Here, we have \(-3y\) and \(-5y\) on separate sides of the equation. To bring them together, add \(5y\) on both sides to keep the equation balanced. It will change the equation to \( 2y+14=0\).

Step 2: Isolate y

Next step is to isolate y on one side of the equation. To do that, subtract 14 on both sides and it will result in the equation changing to \(2y= -14\).

Step 3: Find the value of y

Finally, we need to find the value of y. y can be found by dividing \(-14\) by the coefficient of y, here being \(2\). Hence, after carrying out the operation we get the value of \(y=-7\).

Key Concepts

Equation Balancing

Equation balancing is a fundamental principle when solving equations. It involves maintaining the equality to ensure that the equation is true. Think of the equation like a balanced scale; whatever operation you perform on one side must be done to the other to keep it balanced.

• If you add 5 on one side, you must add 5 on the other.
• Multiplying, dividing, or subtracting must also occur equally on both sides of the equation.

Remember, the main goal of balancing is to simplify the equation while still representing the original equality. Failing to balance can lead to incorrect answers.

Like Terms

Like terms are terms in an equation that have the same variable raised to the same power. They can be simply added or subtracted. Identifying like terms is crucial in simplifying equations.
In the example equation \[-3y + 14 = -5y\],

• The terms \-3y\ and \ -5y\ are like terms because both include the variable \ y\ with the same exponent of 1.
• To simplify, you combine these terms on one side of the equation.

By moving \ -5y\ to the left side through addition, you ensure all \ y\ terms are together, which allows for simpler manipulation of the equation.

Isolation of Variables

Isolation of variables means getting the variable by itself on one side of the equation. This step is key to solving for the unknown. After you've combined like terms, you should have an equation such as \[2y + 14 = 0\].

• The goal is to have the form \ ax = b\, where \ x\ is isolated.
• The first step is to move all numerical values to the opposite side of the variable. Here, subtracting 14 gives \ 2y = -14\.
• Then, divide each term by the coefficient of the variable (in this case, 2) to solve, resulting in \ y = -7\.

Isolation allows us to find the specific value of the variable, offering a clear solution to the equation.