Solving Systems of Linear Equations by Substitution : Videos & Practice Problems

The substitution method solves systems of equations by isolating a variable, then substituting it into another equation to simplify and solve for one variable. For example, given $y=7x-14$, substitute into $2x-y=4$ to find $x$. After solving for $x$, substitute back to find $y$. This method avoids guesswork and ensures accurate solutions, reinforcing understanding of terms, coefficients, and polynomial expressions in standard form.

Solving Systems of Linear Equations - Substitution

In solving systems of equations, the substitution method provides a systematic approach to find the values of variables without the need for graphing. This method is particularly useful when equations are complex or when graphing is impractical. The solution to a system of equations consists of a pair of numbers (x, y) that satisfy both equations simultaneously.

The first step in the substitution method is to identify the easier equation to isolate one of the variables, typically denoted as equation A. For instance, if you have the equation y = 7x - 14, it is already solved for y, making it a suitable choice for equation A. The other equation will be designated as equation B.

Next, you will substitute the expression from equation A into equation B. This means that wherever the variable y appears in equation B, you will replace it with the expression from equation A. For example, if equation B is 2x - y = 4, substituting gives you 2x - (7x - 14) = 4. This substitution simplifies the equation to a single variable, allowing you to solve for x.

After performing the substitution, you will simplify the equation. Continuing with our example, you would distribute the negative sign and combine like terms, resulting in -5x + 14 = 4. Solving this equation leads to x = 2.

With the value of x determined, the next step is to find y by substituting x back into either equation A or B. Using equation A, you would substitute x = 2 into y = 7(2) - 14, which simplifies to y = 0. Thus, the solution to the system of equations is (x, y) = (2, 0).

Finally, it is essential to verify your solution by substituting both values back into the original equations. For equation A, substituting gives 0 = 7(2) - 14, which holds true. For equation B, substituting yields 2(2) - 0 = 4, confirming the solution is correct. This verification step ensures that the values found satisfy both equations, solidifying the accuracy of the substitution method.

Use substitution to solve the following system of linear equations.
$4x+y=1$
$x-y=4$

• A. $\left(-1,3\right)$
• B. $\left(-3,1\right)$
• C. $\left(3,-1\right)$
• D. $\left(1,-3\right)$

Answer: (D)

Use substitution to solve the following system of linear equations.
$4x+2y=7$
$x+5y=4$

• A. $(\frac{1}{2},\frac{3}{2})$
• B. $\left(\frac32,\frac12\right)$
• C. $\left(-\frac23,\frac13\right)$
• D. $\left(-\frac13,\frac23\right)$

Answer: (B)

Here’s what students ask on this topic:

What is the substitution method for solving systems of linear equations?

The substitution method is a technique used to solve systems of linear equations by isolating one variable in one equation and then substituting that expression into the other equation. This process reduces the system to a single equation with one variable, which can be solved more easily. For example, if you have the system $y=7x-14$ and $2x-y=4$, you can substitute the expression for $y$ from the first equation into the second. This eliminates $y$ and allows you to solve for $x$. After finding $x$, substitute it back into one of the original equations to find $y$. This method avoids guesswork and provides a systematic way to find exact solutions.

How do you choose which variable to isolate when using the substitution method?

When using the substitution method, you should choose to isolate the variable that is easiest to solve for in one of the equations. This usually means selecting the variable that already has a coefficient of 1 or -1, or the one that can be isolated with the least amount of algebraic manipulation. For example, if an equation is already written as $y=...$, it is simplest to isolate $y$. Isolating the easier variable simplifies the substitution step and reduces the chance of errors. After isolating the variable, substitute its expression into the other equation to solve for the remaining variable.

Can the substitution method be used for any system of linear equations?

Yes, the substitution method can be used to solve any system of linear equations, whether the system has one solution, infinitely many solutions, or no solution. However, it is most efficient when one of the equations is already solved for one variable or can be easily manipulated to isolate a variable. For more complex systems, other methods like elimination or matrix methods might be more practical. The substitution method is especially useful for systems with two equations and two variables, providing a clear, step-by-step approach to find exact solutions.

How do you check your solution after using the substitution method?

After finding the values of the variables using the substitution method, you should check your solution by substituting these values back into both original equations. This ensures that the solution satisfies both equations. For example, if you found $x=2$ and $y=0$, plug these into each equation. If both equations hold true (i.e., both sides are equal), your solution is correct. This verification step helps catch any mistakes made during the solving process and confirms the accuracy of your answer.

What are the steps involved in solving a system of equations by substitution?

The substitution method involves five main steps: (1) Choose the easiest equation to isolate one variable and label it as equation $a$. (2) Solve equation $a$ for the chosen variable. (3) Substitute this expression into the other equation (equation $b$) to eliminate one variable. (4) Solve the resulting single-variable equation. (5) Substitute the found value back into equation $a$ to find the other variable. Finally, (6) check your solution by plugging both values into the original equations to verify correctness. This systematic approach simplifies solving systems without graphing.