Solving Systems Of Equations: A Step-by-Step Guide

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Solving Systems of Equations: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving deep into the world of solving systems of equations. Specifically, we'll tackle the equations 4x + 6y = 2 and 6x + 5y = 1. This isn't just about finding x and y; it's about understanding the core concepts and techniques that make solving these problems a breeze. So, grab your pencils, and let's get started!

Understanding Systems of Equations

Before we jump into the nitty-gritty, let's make sure we're all on the same page. A system of equations is simply a set of two or more equations that we want to solve simultaneously. The goal? To find the values of the variables (in our case, x and y) that satisfy all the equations in the system. Think of it like a treasure hunt where the treasure (the solution) must be hidden at a location that satisfies all clues (equations). There are several ways to solve these systems, and we'll explore some of the most common methods.

Why Are Systems of Equations Important?

You might be wondering, "Why do I need to learn this?" Well, systems of equations are incredibly useful in real-world scenarios. They can model situations like: finding the break-even point in a business (where the costs equal the revenue), calculating the intersection of two paths, or even predicting population growth. They're fundamental in fields like physics, engineering, economics, and computer science. Basically, if you want to model and understand relationships between multiple variables, systems of equations are your best friends.

Methods for Solving Systems of Equations

There are several methods for solving systems of equations, each with its own advantages. The most common include:

  • Substitution: Solving one equation for one variable and then substituting that expression into the other equation.
  • Elimination: Manipulating the equations (usually by multiplying them by constants) so that when you add or subtract them, one of the variables is eliminated.
  • Graphing: Plotting the equations on a graph and finding the point(s) where they intersect.

We will explore the Substitution and Elimination Methods for these equations to provide you with the most useful approaches.

Solving 4x + 6y = 2 and 6x + 5y = 1 Using the Substitution Method

Alright, let's put our knowledge to the test. Let's solve the equations 4x + 6y = 2 and 6x + 5y = 1 using the substitution method. This method works like this: we isolate one variable in one equation and then substitute that expression into the other equation.

Step-by-Step Guide to Substitution

  1. Isolate a Variable: Choose one of the equations and solve for one of the variables. Let's start with the first equation 4x + 6y = 2. It looks easier to solve for x. So we'll rearrange this equation to isolate x:

    4x = 2 - 6y x = (2 - 6y) / 4 x = 1/2 - (3/2)y

  2. Substitute: Now that we have an expression for x, substitute it into the second equation, 6x + 5y = 1:

    6 * (1/2 - (3/2)y) + 5y = 1

  3. Solve for the Remaining Variable: Simplify and solve for y:

    3 - 9y + 5y = 1 -4y = -2 y = 1/2

  4. Solve for the Other Variable: Now that we know y = 1/2, plug this value back into either of the original equations (or the expression we found for x) to solve for x. Let's use x = 1/2 - (3/2)y:

    x = 1/2 - (3/2) * (1/2) x = 1/2 - 3/4 x = -1/4

  5. Solution: So, the solution to the system of equations is x = -1/4 and y = 1/2. We can write this as an ordered pair (-1/4, 1/2). This represents the point where the two lines represented by the equations intersect.

Solving 4x + 6y = 2 and 6x + 5y = 1 Using the Elimination Method

Now, let's tackle the same system of equations, 4x + 6y = 2 and 6x + 5y = 1, using the elimination method. The elimination method involves manipulating the equations so that when you add or subtract them, one of the variables cancels out. This often involves multiplying one or both equations by a constant.

Step-by-Step Guide to Elimination

  1. Prepare the Equations: We need to make the coefficients of either x or y opposites. Let's aim to eliminate x. Multiply the first equation by 3 and the second equation by -2. This way, the coefficients of x will be 12 and -12, respectively.

    • Equation 1 (multiplied by 3): 12x + 18y = 6
    • Equation 2 (multiplied by -2): -12x - 10y = -2

  2. Eliminate a Variable: Add the modified equations together. The x terms will cancel out:

    (12x + 18y) + (-12x - 10y) = 6 + (-2) 8y = 4

  3. Solve for the Remaining Variable: Solve for y:

    y = 4 / 8 y = 1/2

  4. Solve for the Other Variable: Substitute the value of y back into either of the original equations to solve for x. Let's use the first original equation 4x + 6y = 2:

    4x + 6 * (1/2) = 2 4x + 3 = 2 4x = -1 x = -1/4

  5. Solution: The solution is again x = -1/4 and y = 1/2, or the ordered pair (-1/4, 1/2). This confirms that both methods lead to the same solution.

Graphing the Equations

Let's visualize the equations graphically. The graph of a linear equation like 4x + 6y = 2 is a straight line. The solution to the system of equations is the point where these two lines intersect. Let's sketch it. For that, we need to convert the equations in the slope-intercept form, y = mx + c.

Convert to Slope-Intercept Form

  1. Equation 1: 4x + 6y = 2

    6y = -4x + 2 y = (-4/6)x + 2/6 y = (-2/3)x + 1/3

  2. Equation 2: 6x + 5y = 1

    5y = -6x + 1 y = (-6/5)x + 1/5

Plotting the Lines

Now, for each equation, we have the slope (m) and the y-intercept (c).

  • Equation 1: Slope = -2/3, y-intercept = 1/3
  • Equation 2: Slope = -6/5, y-intercept = 1/5

To graph each line:

  1. Plot the y-intercept: Mark the point where the line crosses the y-axis (1/3 for the first equation and 1/5 for the second).
  2. Use the slope: From the y-intercept, use the slope to find another point. For example, for the first equation, the slope is -2/3. This means that from the y-intercept, you go down 2 units and right 3 units. You can find another point in the line.
  3. Draw the line: Draw a straight line through the two points you've found for each equation. The point where the two lines intersect is the solution to the system of equations. In our case, this point should be very close to (-1/4, 1/2) which we already determined. The graphing process may not always produce such accuracy but it can still be used to determine the solution.

Discussion and Conclusion

As you can see, solving systems of equations is not that intimidating after all! We explored two powerful methods – substitution and elimination – to find the solution, and we also graphed the equations to visualize the solution. Each method has its pros and cons. The substitution method can be more straightforward when one of the variables is already isolated or easily isolated in one of the equations. The elimination method is particularly useful when the coefficients of one of the variables are easily made opposites. The choice of which method to use often depends on the specific equations you're working with and your personal preference.

Key Takeaways

  • Systems of equations are sets of equations solved simultaneously.
  • The solution to a system of equations is the set of values for the variables that satisfy all equations.
  • The substitution method involves solving for one variable and substituting.
  • The elimination method involves manipulating equations to eliminate a variable.
  • Graphing the equations visually represents the solution as the point of intersection.

Final Thoughts

Practice is key! The more you practice solving systems of equations, the more comfortable you'll become. Try solving different systems of equations, experiment with different methods, and don't be afraid to make mistakes – that's how we learn. And remember, understanding the underlying concepts is more important than memorizing formulas. Now go out there and conquer those equations, and remember that with a little practice, it's totally achievable!