What Is The Solution Of The System Of Equations

What is the Solution of the System of Equations?

Imagine you’re managing a small coffee shop. Worth adding: you sell two types of drinks: espressos and lattes. Also, the answer—25 espressos and 25 lattes—is the solution. In your coffee shop, that means finding the exact number of espressos (let's say x) and lattes (y) that satisfy both the total drink count (x + y = 50) and the total revenue (8x + 10y = 420). How many of each drink did you sell? Day to day, the solution of the system of equations is the specific set of values for the variables that makes every equation in the system true simultaneously. So naturally, this real-world puzzle isn't solved by a single equation but by a system of equations. On a busy morning, you sell a total of 50 drinks and collect $420. An espresso costs $8, and a latte costs $10. Understanding how to find these solutions is a foundational skill in algebra with powerful applications in science, engineering, economics, and everyday problem-solving.

Understanding the Core Concept: What Constitutes a Solution?

A system of equations is simply a collection of two or more equations with the same set of variables. The goal is to find the values for those variables that satisfy all equations at once. For a system of two linear equations in two variables, the solution corresponds to the point of intersection of the two lines when graphed on a coordinate plane. This intersection point has coordinates (x, y) that work in both original equations.

Systems can have:

1. Infinitely Many Solutions: The lines are coincident (the same line). On top of that, 2. This is an inconsistent system. This is the most common case for independent equations.
2. So No Solution: The lines are parallel and never meet. One Unique Solution: The lines intersect at exactly one point. Every point on that line is a solution, making the system dependent.

The method you choose to find this solution depends on the system's form and your objective—whether it's a quick algebraic answer or a visual understanding.

Primary Algebraic Methods for Finding the Solution

1. The Substitution Method

This method is highly effective when one equation is already solved for one variable or can be easily manipulated to do so.

• Step 1: Solve one of the equations for one variable. Take this: from x + y = 50, solve for y: y = 50 - x.
• Step 2: Substitute this expression into the other equation. Replace y in 8x + 10y = 420 with (50 - x), giving 8x + 10(50 - x) = 420.
• Step 3: Solve the resulting single-variable equation. 8x + 500 - 10x = 420 simplifies to -2x = -80, so x = 40.
• Step 4: Substitute this value back into the expression from Step 1 to find the other variable. y = 50 - 40 = 10.
• Step 5: Verify the solution by plugging x=40 and y=10 into both original equations. This crucial step catches algebraic errors.

Best for: Systems where one variable has a coefficient of 1 or -1, making isolation simple.

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2. The Elimination Method (Addition/Subtraction Method)

This method is often more efficient when coefficients are not 1. The goal is to add or subtract the equations to eliminate one variable.

• Step 1: Align the equations in standard form (Ax + By = C). Your system: x + y = 50 and 8x + 10y = 420.
• Step 2: Manipulate one or both equations by multiplying by constants so that the coefficients of one variable are opposites. To eliminate y, multiply the first equation by -10: -10x - 10y = -500.
• Step 3: Add the modified first equation to the second equation. (-10x - 10y) + (8x + 10y) = -500 + 420 simplifies to -2x = -80.
• Step 4: Solve for the remaining variable (x = 40).
• Step 5: Substitute back to find the other variable (y = 10).
• Step 6: Verify in