Solving for y in the Equation 2x + 3y = 6
Introduction
When you encounter a linear equation such as 2x + 3y = 6, the goal of *solving for y is to rewrite the equation so that y appears alone on one side, expressed as a function of x. Consider this: this process, called isolating the variable, is a fundamental skill in algebra and serves as the foundation for graphing lines, analyzing relationships between variables, and solving more complex systems of equations. In this article we will walk through the step‑by‑step manipulation required to solve 2x + 3y = 6 for y, explore why each step works, and provide practical examples that reinforce the technique No workaround needed..
Step‑by‑Step Procedure
1. Write the original equation
[ 2x + 3y = 6 ]
The equation is already in its simplest form, with all terms on one side and the constant on the other.
2. Subtract 2x from both sides
To move the term containing x out of the way, we perform the same operation on both sides of the equality. This preserves the balance of the equation:
[ 2x + 3y - 2x = 6 - 2x ]
Simplifying gives:
[ 3y = 6 - 2x ]
Why this works: Adding or subtracting the same quantity from both sides of an equation does not change the solution set; it merely re‑positions the terms Most people skip this — try not to..
3. Divide every term by 3
Now y is multiplied by 3. To isolate y, divide both sides by 3:
[ \frac{3y}{3} = \frac{6 - 2x}{3} ]
Which simplifies to:
[ y = \frac{6}{3} - \frac{2x}{3} ]
4. Simplify the fractions
[ y = 2 - \frac{2}{3}x ]
It is often helpful to write the equation in slope‑intercept form (y = mx + b). Re‑ordering the terms gives:
[ \boxed{y = -\frac{2}{3}x + 2} ]
This final expression tells us that for any chosen value of x, we can compute y directly, and it also reveals the line’s slope (-\frac{2}{3}) and y‑intercept at ((0,2)).
Why the Process Matters
Understanding Linear Relationships
A linear equation in two variables describes a straight line when plotted on the coordinate plane. By solving for y, we convert the equation into a function (y = f(x)). This functional form is essential for:
- Graphing: You can quickly identify the slope and intercept, allowing you to draw the line without plotting multiple points.
- Predictive Modeling: In fields such as economics, physics, and engineering, the function form lets you predict the value of one variable based on another.
Preparing for Advanced Topics
Mastering variable isolation is a prerequisite for:
- Systems of equations: Solving for one variable often lets you substitute back into another equation.
- Inequalities: The same algebraic steps apply, though you must remember to reverse the inequality sign when multiplying or dividing by a negative number.
- Calculus: Implicit differentiation and solving for derivatives frequently require isolating a variable first.
Practical Examples
Example 1: Direct Substitution
Suppose (x = 3). Using the solved equation (y = -\frac{2}{3}x + 2):
[ y = -\frac{2}{3}(3) + 2 = -2 + 2 = 0 ]
Thus the point ((3,0)) lies on the line.
Example 2: Solving for a Specific y Value
If you need the x‑coordinate when (y = -1):
[ -1 = -\frac{2}{3}x + 2 \ -1 - 2 = -\frac{2}{3}x \ -3 = -\frac{2}{3}x \ x = \frac{-3}{-2/3} = \frac{-3 \times 3}{-2} = \frac{9}{2} = 4.5 ]
So the point ((4.5, -1)) satisfies the original equation Not complicated — just consistent. Nothing fancy..
Example 3: Real‑World Context
A small business sells x units of product A and y units of product B. Their revenue model is (2x + 3y = 6) (in thousands of dollars). Solving for y shows how many units of product B can be sold once the quantity of product A is known:
[ y = -\frac{2}{3}x + 2 ]
If the company plans to sell 6 units of product A ((x = 6)):
[ y = -\frac{2}{3}(6) + 2 = -4 + 2 = -2 ]
A negative y indicates that selling 6 units of product A would exceed the revenue constraint, highlighting the need to adjust production plans Which is the point..
Common Pitfalls and How to Avoid Them
| Mistake | Why It Happens | How to Fix It |
|---|---|---|
| Forgetting to divide every term | Students sometimes divide only the left side or only the first term on the right. | Always apply the operation to all terms on both sides. Now, |
| Sign errors when moving terms | Subtracting (2x) can be confused with adding (-2x). | Write the step explicitly: (2x + 3y - 2x = 6 - 2x). |
| Incorrect slope‑intercept ordering | Writing (y = 2 - \frac{2}{3}x) is mathematically correct but less conventional. | Re‑order to (y = -\frac{2}{3}x + 2) for clarity. So |
| Misapplying the distributive property | If the equation were (2(x + 3y) = 6), students might forget to multiply both x and 3y. | Always expand parentheses before isolating variables. |
Frequently Asked Questions (FAQ)
Q1: What if the coefficient of y is negative?
A: The same steps apply. Take this: if the equation were (2x - 3y = 6), you would add (2x) to both sides, then divide by (-3). The sign of the coefficient simply carries through to the final slope Surprisingly effective..
Q2: Can I solve for y if there are fractions in the original equation?
A: Yes. Clear fractions first by multiplying every term by the least common denominator, then follow the isolation steps Less friction, more output..
Q3: Does solving for y change the solution set of the original equation?
A: No. Algebraic manipulations that maintain equality (adding, subtracting, multiplying, or dividing by non‑
