Solving Systems Of Equations 3x + 6y = 12 And X + 5y = 7

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Hey everyone! Are you struggling with systems of equations? Don't worry, you're not alone! Many students find these a bit tricky at first, but with a little practice, you'll be solving them like a pro. In this article, we're going to break down a classic example step-by-step: solving the system of equations 3x + 6y = 12 and x + 5y = 7. We'll explore different methods and make sure you understand the whys behind the hows. So, grab your pencils and let's dive in!

Understanding Systems of Equations

Before we jump into solving, let's make sure we're all on the same page about what a system of equations actually is. Simply put, a system of equations is a set of two or more equations that contain the same variables. The goal is to find the values for those variables that make all the equations in the system true simultaneously. Think of it like a puzzle where you need to find the perfect pieces (variable values) that fit into multiple slots (equations).

In our case, we have two equations:

  1. 3x + 6y = 12
  2. x + 5y = 7

We need to find the values of x and y that satisfy both of these equations. These values represent the point where the lines represented by these equations intersect on a graph. There are several methods to solve such systems, and we’ll explore the most common ones here: the substitution method, the elimination method, and even a graphical approach to give you a visual understanding. Understanding these methods will not only help you solve this specific problem but will also equip you with the tools to tackle a wide range of similar problems. Remember, mathematics is not just about finding the right answer; it's about understanding the process and developing critical thinking skills. So, let's embark on this journey together and unravel the mysteries of systems of equations!

Why are Systems of Equations Important?

You might be wondering, why bother learning this stuff? Well, systems of equations are incredibly useful in the real world! They pop up in various fields, from science and engineering to economics and computer science. For instance, they can be used to model supply and demand in economics, to analyze circuits in electrical engineering, or even to optimize logistics in transportation. Learning how to solve these systems opens doors to understanding and solving a multitude of practical problems.

Imagine you're planning a road trip and need to figure out how much gas to buy. You know your car's fuel efficiency and the total distance, but gas prices vary along the route. You can set up a system of equations to minimize your fuel costs! Or, suppose you're a chef trying to create a new recipe. You know the desired nutritional content (calories, protein, etc.) and the nutritional information for various ingredients. Systems of equations can help you determine the exact quantities of each ingredient to use.

In fact, systems of equations are a fundamental tool in any field that involves mathematical modeling. They allow us to represent complex relationships between different variables and find solutions that satisfy multiple constraints. So, by mastering this concept, you're not just learning a mathematical technique; you're developing a powerful problem-solving skill that will serve you well in many areas of life.

Method 1: The Substitution Method

The substitution method is a powerful technique for solving systems of equations. The basic idea is to solve one equation for one variable and then substitute that expression into the other equation. This eliminates one variable, leaving you with a single equation in one variable, which is much easier to solve. Once you've found the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable.

Let's apply this method to our system:

  1. 3x + 6y = 12
  2. x + 5y = 7

Step-by-Step Guide

  1. Solve one equation for one variable: Looking at our equations, the second equation (x + 5y = 7) seems easier to solve for x. Let's isolate x:

    x = 7 - 5y

    Great! We've now expressed x in terms of y. This is a crucial step because we can now replace x in the other equation with this expression.

  2. Substitute: Now, we'll substitute this expression for x (7 - 5y) into the first equation (3x + 6y = 12):

    3(7 - 5y) + 6y = 12

    Notice how we've replaced x with (7 - 5y). This is the heart of the substitution method! We've transformed the first equation into an equation with only one variable, y. This makes it much easier to solve.

  3. Solve for the remaining variable: Let's simplify and solve for y:

    21 - 15y + 6y = 12 -9y = -9 y = 1

    Excellent! We've found the value of y: y = 1. This is one piece of the puzzle. Now we need to find the value of x.

  4. Substitute back: We'll substitute the value of y (1) back into either of the original equations or the expression we found for x. Let's use the expression x = 7 - 5y:

    x = 7 - 5(1) x = 7 - 5 x = 2

    We've found the value of x: x = 2.

  5. Check your solution: It's always a good idea to check your solution by plugging the values of x and y back into the original equations to make sure they hold true:

    • Equation 1: 3(2) + 6(1) = 6 + 6 = 12 (Correct!)
    • Equation 2: 2 + 5(1) = 2 + 5 = 7 (Correct!)

    Since our solution satisfies both equations, we know we've found the correct answer.

  6. State the solution: The solution to the system of equations is x = 2 and y = 1. We can also write this as an ordered pair: (2, 1). This ordered pair represents the point where the two lines intersect on a graph.

Method 2: The Elimination Method

The elimination method, also known as the addition method, is another powerful technique for solving systems of equations. Instead of solving for one variable and substituting, this method focuses on eliminating one of the variables by adding or subtracting the equations. The key is to manipulate the equations so that the coefficients of one variable are opposites (e.g., 3x and -3x). When you add the equations, that variable will be eliminated, leaving you with a single equation in one variable.

Let's tackle our system again using this method:

  1. 3x + 6y = 12
  2. x + 5y = 7

Step-by-Step Guide

  1. Multiply equations to match coefficients: Our goal is to make the coefficients of either x or y opposites. Let's choose to eliminate x. We can multiply the second equation by -3 so that the x coefficient becomes -3:

    • Equation 1: 3x + 6y = 12
    • Equation 2 (multiplied by -3): -3x - 15y = -21

    Now we have 3x in the first equation and -3x in the second equation – perfect for elimination!

  2. Add the equations: Add the two equations together:

    (3x + 6y) + (-3x - 15y) = 12 + (-21) -9y = -9

    Notice how the x terms canceled out! We're left with a single equation in y.

  3. Solve for the remaining variable: Solve for y:

    y = 1

    We've found y = 1, just like with the substitution method.

  4. Substitute back: Substitute the value of y (1) back into either of the original equations to solve for x. Let's use the second equation:

    x + 5(1) = 7 x + 5 = 7 x = 2

    We've found x = 2.

  5. Check your solution: Again, let's check our solution in both original equations:

    • Equation 1: 3(2) + 6(1) = 6 + 6 = 12 (Correct!)
    • Equation 2: 2 + 5(1) = 2 + 5 = 7 (Correct!)

  6. State the solution: The solution is x = 2 and y = 1, or the ordered pair (2, 1).

Method 3: The Graphical Method (Visualizing the Solution)

While the substitution and elimination methods are algebraic approaches, the graphical method provides a visual way to understand and solve systems of equations. Each equation in the system represents a line on a graph. The solution to the system is the point where these lines intersect. If the lines don't intersect (they are parallel), there's no solution. If the lines are the same, there are infinitely many solutions.

Let's graph our equations:

  1. 3x + 6y = 12
  2. x + 5y = 7

Step-by-Step Guide

  1. Rewrite equations in slope-intercept form: To easily graph the lines, it's helpful to rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

    • Equation 1: 3x + 6y = 12 => 6y = -3x + 12 => y = (-1/2)x + 2
    • Equation 2: x + 5y = 7 => 5y = -x + 7 => y = (-1/5)x + 7/5

    Now we can easily identify the slopes and y-intercepts for each line.

  2. Plot the lines:

    • For Equation 1 (y = (-1/2)x + 2), the y-intercept is 2, and the slope is -1/2. This means we start at the point (0, 2) and then go down 1 unit and right 2 units to find another point on the line. Draw the line through these points.
    • For Equation 2 (y = (-1/5)x + 7/5), the y-intercept is 7/5 (or 1.4), and the slope is -1/5. Start at (0, 1.4) and then go down 1 unit and right 5 units to find another point. Draw the line.

  3. Find the intersection point: The point where the two lines intersect is the solution to the system of equations. If you graph the lines accurately, you'll see that they intersect at the point (2, 1).

  4. State the solution: The graphical method confirms our previous results. The solution is x = 2 and y = 1, or the ordered pair (2, 1).

Choosing the Best Method

Now that we've explored three different methods, you might be wondering which one is the