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

Let's dive into solving a system of equations! Systems of equations pop up everywhere, from simple math problems to complex real-world scenarios. Here, we're going to tackle a specific system and break down each step to make sure you understand exactly how to solve it. We'll focus on the substitution method, which is super handy when one of the equations already has a variable isolated. So, grab a pencil, and let's get started!

The System of Equations

We're given the following system:

$$\left\{ \begin{array}{l} x = 6y - 14 \\ -x + 9y = 17 \end{array} \right.$$

Our mission is to find the values of x and y that satisfy both equations simultaneously. In other words, we need to find the point where these two equations intersect if we were to graph them.

Step-by-Step Solution

Step 1: Identify the Best Method

When solving systems of equations, you have a few options, including substitution, elimination, and graphing. In this case, the substitution method is the most straightforward because the first equation is already solved for x. This means we can easily substitute the expression for x from the first equation into the second equation. This simplifies the problem and allows us to solve for y directly.

Step 2: Substitute

Now, let’s substitute the expression for x from the first equation, which is x = 6y - 14, into the second equation:

-x + 9y = 17

becomes

-(6y - 14) + 9y = 17

See what we did there? We replaced x in the second equation with the entire expression (6y - 14) from the first equation. Make sure to put the expression in parentheses to avoid any sign errors. This step is crucial because it reduces the system from two variables to a single variable, which we can then easily solve.

Step 3: Simplify and Solve for y

Next, we simplify the equation and solve for y:

-(6y - 14) + 9y = 17

Distribute the negative sign:

-6y + 14 + 9y = 17

Combine like terms:

3y + 14 = 17

Subtract 14 from both sides:

3y = 3

Divide by 3:

y = 1

So, we've found that y = 1. This is a big step! We now know the y-coordinate of the solution. Understanding each of these algebraic manipulations is essential. Distributing the negative sign correctly, combining like terms, and isolating the variable are all fundamental skills in algebra. Make sure you practice these steps to become more comfortable with them.

Step 4: Solve for x

Now that we know y = 1, we can plug this value back into either of the original equations to solve for x. It’s usually easiest to use the equation that's already solved for x, which is:

x = 6y - 14

Substitute y = 1:

x = 6(1) - 14

x = 6 - 14

x = -8

So, x = -8. We now have both the x and y values that satisfy the system of equations.

Step 5: Check Your Solution

To make sure we didn't make any mistakes, let's check our solution by plugging x = -8 and y = 1 into both original equations:

Equation 1: x = 6y - 14

-8 = 6(1) - 14

-8 = 6 - 14

-8 = -8 (Correct!)

Equation 2: -x + 9y = 17

-(-8) + 9(1) = 17

8 + 9 = 17

17 = 17 (Correct!)

Since our solution satisfies both equations, we know we've found the correct answer. Checking your work is always a good habit to develop in mathematics. It helps you catch any errors and ensures that your solution is accurate.

Final Answer

The solution to the system of equations is x = -8 and y = 1. We can write this as an ordered pair: (-8, 1).

Explanation of the Solution

Understanding the Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable and allows you to solve for the remaining variable. Once you find the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. This method is particularly useful when one of the equations is already solved for a variable or can be easily solved for a variable.

Why This Works

The reason this method works is that we are essentially finding the point where the two equations intersect. When we substitute one equation into the other, we are finding the values of x and y that satisfy both equations simultaneously. This point of intersection is the solution to the system of equations. Graphically, the solution (-8, 1) represents the coordinates where the two lines defined by the equations x = 6y - 14 and -x + 9y = 17 intersect on the coordinate plane.

Additional Tips and Tricks

When to Use Substitution

Substitution is most effective when one of the equations is already solved for a variable or can be easily solved for a variable. If neither equation is easily solved for a variable, you might consider using the elimination method.

Common Mistakes to Avoid

• Sign Errors: Be very careful with negative signs, especially when distributing. A simple sign error can throw off the entire solution.
• Incorrect Substitution: Make sure you are substituting the correct expression into the correct equation. Double-check your work to avoid this mistake.
• Forgetting to Check: Always check your solution by plugging the values of x and y back into the original equations. This will help you catch any errors and ensure that your solution is correct.

Practice Problems

To master the substitution method, try solving the following systems of equations:

1. y = 2x + 1 3x + 2y = 16

2. x = 3y - 5 2x - 5y = -8

3. a = 4b + 2 3a - 10b = 8

Solving these problems will give you more confidence and skill in using the substitution method.

Conclusion

Solving systems of equations is a fundamental skill in algebra. By understanding the substitution method and practicing regularly, you can confidently solve a wide range of problems. Remember to take your time, double-check your work, and don't be afraid to ask for help if you get stuck. With practice, you'll become a pro at solving systems of equations!

For further learning and practice, check out resources like Khan Academy's Systems of Equations section.