Solving Systems Of Equations Find The Solution To Y = 6x - 11 -2x - 3y = -7

In the realm of mathematics, particularly in algebra, solving systems of equations is a fundamental skill. These systems represent scenarios where two or more equations share common variables, and the goal is to find the values of these variables that satisfy all equations simultaneously. In this article, we will explore a specific system of equations and delve into the process of finding its solution. Let’s consider the system:

\begin{cases}
y = 6x - 11 \\
-2x - 3y = -7
\end{cases}

We are presented with four potential solutions:

A. $\left(-\frac{13}{10},-\frac{94}{5}\right)$

B. $\left(-\frac{1}{2},-14\right)$

C. $(1,-5)$

D. $(2,1)$

Our mission is to determine which of these options, if any, correctly solves the system. To accomplish this, we'll employ the method of substitution, a powerful technique for solving systems of equations. This involves solving one equation for one variable and substituting that expression into the other equation. This systematic approach allows us to reduce the system to a single equation with one variable, which can then be easily solved. The resulting value is then substituted back into either of the original equations to solve for the other variable. This comprehensive process ensures that the solution we obtain satisfies both equations in the system.

Method of Substitution

The substitution method is a powerful algebraic technique used to solve systems of equations. It shines when one equation is already solved for one variable, or can be easily manipulated to that form. In such scenarios, substitution provides a streamlined path to finding the solution. Let’s break down the steps involved:

1. Isolate a Variable: Begin by selecting one of the equations and solving it for one variable. This means expressing one variable in terms of the other. The choice of which equation and which variable to isolate often depends on which option appears easiest to manipulate. For instance, if one equation already has a variable with a coefficient of 1, isolating that variable is usually the most straightforward approach. In our system, the first equation, y = 6x - 11, is already solved for y, making it an ideal candidate for substitution.
2. Substitute: Once a variable is isolated, substitute the expression you found in step 1 into the other equation. This effectively replaces the isolated variable in the second equation with an equivalent expression involving the other variable. The result is a new equation that contains only one variable, making it solvable using standard algebraic techniques. This step is crucial as it reduces the system of two equations into a single equation, which simplifies the solving process significantly.
3. Solve for the Remaining Variable: With the substitution made, you now have a single equation with one unknown variable. Solve this equation using algebraic methods. This might involve simplifying the equation by combining like terms, distributing coefficients, or applying inverse operations to isolate the variable. The goal is to find the numerical value of this variable that satisfies the equation. Once you've found this value, you're halfway to solving the system.
4. Back-Substitute: Once you've solved for one variable, substitute its value back into either of the original equations (or the expression you found in step 1) to solve for the other variable. This step allows you to determine the value of the second variable, completing the solution process. Choose the equation that appears easier to work with to minimize the complexity of the calculations. By substituting the known value back into an equation, you create a new equation with only one unknown, which can be easily solved.
5. Verify the Solution: As a final step, it's essential to check your solution by substituting the values you found for both variables into both of the original equations. This ensures that your solution satisfies both equations simultaneously. If the values satisfy both equations, you've found the correct solution to the system. If not, it indicates an error in your calculations, and you'll need to review your steps to identify and correct the mistake. This verification step is a crucial safeguard against errors and ensures the accuracy of your solution.

By following these steps diligently, the method of substitution can be applied to a wide range of systems of equations, providing a reliable and systematic approach to finding solutions. Its strength lies in its ability to simplify the problem by reducing the number of variables, making complex systems solvable through basic algebraic manipulations.

Applying the Substitution Method

In our given system:

\begin{cases}
y = 6x - 11 \\
-2x - 3y = -7
\end{cases}

We can see that the first equation is already solved for y. Therefore, we can substitute the expression 6x - 11 for y in the second equation:

-2x - 3(6x - 11) = -7

Now, let’s simplify and solve for x:

-2x - 18x + 33 = -7

Combine like terms:

-20x + 33 = -7

Subtract 33 from both sides:

-20x = -40

Divide by -20:

x = 2

Now that we have the value of x, we can substitute it back into either of the original equations to find y. Let’s use the first equation:

y = 6(2) - 11

y = 12 - 11

y = 1

So, we have found that x = 2 and y = 1. This gives us the solution (2, 1).

Verifying the Solution

To ensure our solution is correct, we must verify it by substituting the values x = 2 and y = 1 into both original equations:

For the first equation:

y = 6x - 11

1 = 6(2) - 11

1 = 12 - 11

1 = 1  (True)

For the second equation:

-2x - 3y = -7

-2(2) - 3(1) = -7

-4 - 3 = -7

-7 = -7  (True)

Since the values x = 2 and y = 1 satisfy both equations, our solution is correct.

Conclusion

In this article, we successfully solved the system of equations:

\begin{cases}
y = 6x - 11 \\
-2x - 3y = -7
\end{cases}

using the method of substitution. We found that the solution is (2, 1), which corresponds to option D.

Solving systems of equations is a crucial skill in algebra, with applications in various fields, including science, engineering, and economics. The method of substitution provides a systematic approach to finding solutions, making it a valuable tool for any mathematics student or professional. By mastering this technique, one can confidently tackle a wide range of problems involving multiple equations and variables. The ability to solve these systems opens doors to understanding and modeling complex relationships in the world around us.

What is the solution to the system of equations: y = 6x - 11 and -2x - 3y = -7?

Solving Systems of Equations Find the Solution to y = 6x - 11 -2x - 3y = -7