Choosing the Easiest Variable to Solve For

Oct 13, 2025 by windailys.com 145 views

Introduction

Solving linear systems of equations is a fundamental concept in mathematics, and there are several methods to approach this problem. One of the most common methods is substitution, which involves solving one equation for a variable and then substituting that expression into the other equation. In this article, we will explore how to use substitution to solve a linear system of equations.

Understanding the Problem

The problem presented to us is a system of two linear equations with two variables, x and y. The equations are:

\begin{cases} x + 6y = 25 \\ 2x + 2y = 9 \end{cases}

Our goal is to solve for the values of x and y that satisfy both equations.

Choosing the Easiest Variable to Solve For

When using substitution, it is essential to choose the variable that is easiest to solve for. In this case, we need to determine which variable is the easiest to solve for.

To do this, we can examine the coefficients of the variables in both equations. The coefficient of a variable is the number that multiplies the variable in the equation.

In the first equation, the coefficient of x is 1, and the coefficient of y is 6. In the second equation, the coefficient of x is 2, and the coefficient of y is 2.

We can see that the coefficient of y in the first equation is larger than the coefficient of y in the second equation. This means that it is easier to solve for y in the first equation.

Solving for y in the First Equation

To solve for y in the first equation, we can isolate y by subtracting 6y from both sides of the equation.

x + 6y = 25

Subtracting 6y from both sides gives us:

x = 25 - 6y

Now, we can solve for y by dividing both sides of the equation by -6.

y = \frac{25 - x}{6}

Substituting the Expression for y into the Second Equation

Now that we have an expression for y in terms of x, we can substitute this expression into the second equation.

The second equation is:

2x + 2y = 9

Substituting the expression for y into this equation gives us:

2x + 2\left(\frac{25 - x}{6}\right) = 9

Simplifying the Equation

To simplify the equation, we can multiply both sides by 6 to eliminate the fraction.

12x + 2(25 - x) = 54

Expanding the equation gives us:

12x + 50 - 2x = 54

Combining like terms gives us:

10x + 50 = 54

Subtracting 50 from both sides gives us:

10x = 4

Dividing both sides by 10 gives us:

x = \frac{4}{10}

Simplifying the fraction gives us:

x = \frac{2}{5}

Finding the Value of y

Now that we have the value of x, we can substitute it into the expression for y that we found earlier.

y = \frac{25 - x}{6}

Substituting x = 2/5 into this equation gives us:

y = \frac{25 - \frac{2}{5}}{6}

Simplifying the fraction gives us:

y = \frac{125 - 2}{30}

Simplifying further gives us:

y = \frac{123}{30}

Simplifying the fraction gives us:

y = \frac{41}{10}

Conclusion

In this article, we used substitution to solve a linear system of equations. We chose the variable that was easiest to solve for, which was y in the first equation. We then substituted the expression for y into the second equation and solved for x. Finally, we found the value of y by substituting the value of x into the expression for y.

By following these steps, we were able to solve the linear system of equations using substitution. This method is a powerful tool for solving systems of equations, and it can be used to solve a wide range of problems in mathematics and other fields.

References

Q: What is the substitution method?

A: The substitution method is a technique used to solve linear systems of equations by solving one equation for a variable and then substituting that expression into the other equation.

Q: Why is it easier to solve for y in the first equation?

A: It is easier to solve for y in the first equation because the coefficient of y in the first equation is larger than the coefficient of y in the second equation. This makes it simpler to isolate y and solve for its value.

Q: How do I choose the variable to solve for first?

A: To choose the variable to solve for first, you should examine the coefficients of the variables in both equations. The variable with the larger coefficient in the first equation is usually the easiest to solve for.

Q: What if I have a system of three or more equations?

A: If you have a system of three or more equations, you can use the substitution method to solve for one variable at a time. However, this can become more complicated and may require the use of other methods, such as elimination or graphing.

Q: Can I use the substitution method to solve a system of equations with fractions?

A: Yes, you can use the substitution method to solve a system of equations with fractions. However, you will need to simplify the fractions and perform the necessary operations to isolate the variable.

Q: What if I get stuck or make a mistake while using the substitution method?

A: If you get stuck or make a mistake while using the substitution method, don't worry! You can always go back and re-examine your work. If you are still having trouble, try using a different method, such as elimination or graphing.

Q: Is the substitution method the only way to solve linear systems of equations?

A: No, the substitution method is not the only way to solve linear systems of equations. There are several other methods, including elimination, graphing, and matrices. Each method has its own strengths and weaknesses, and the choice of method will depend on the specific problem and the level of difficulty.

Q: Can I use the substitution method to solve non-linear systems of equations?

A: No, the substitution method is typically used to solve linear systems of equations. Non-linear systems of equations require more advanced techniques, such as numerical methods or algebraic manipulations.

Q: Are there any online resources or tools that can help me learn the substitution method?

A: Yes, there are many online resources and tools that can help you learn the substitution method. Some popular resources include:

Khan Academy: Linear Systems of Equations
Mathway: Substitution Method
Wolfram Alpha: Linear Systems of Equations

Conclusion

In this article, we have answered some of the most frequently asked questions about solving linear systems of equations using substitution. We have covered topics such as choosing the variable to solve for first, using the substitution method with fractions, and getting stuck or making mistakes. We have also discussed the limitations of the substitution method and provided some online resources for further learning.

By understanding the substitution method and its applications, you can become a more confident and proficient problem-solver in mathematics and other fields.