The method of substitution works particularly well if one or both of the equations has an isolated
variable on one side of the equation.

Question

The method of substitution works particularly well if one or both of the equations has an isolated variable on one side of the equation.

UpStudy AI · Accepted Answer

Yes, the method of substitution is a powerful technique for solving systems of equations, especially when one of the equations has a variable isolated. Here's how the method works step by step:

1. **Identify the Equations**: Start with a system of equations. For example:
   $$
   \begin{align*}
   1) & \quad y = 2x + 3 \
   2) & \quad 3x + 4y = 10
   \end{align*}
   $$

2. **Substitute the Isolated Variable**: Since the first equation has $$y$$ isolated, substitute $$y$$ in the second equation:
   $$
   3x + 4(2x + 3) = 10
   $$

3. **Simplify the Equation**: Distribute and combine like terms:
   $$
   3x + 8x + 12 = 10
   $$
   $$
   11x + 12 = 10
   $$

4. **Solve for the Variable**: Isolate $$x$$:
   $$
   11x = 10 - 12
   $$
   $$
   11x = -2
   $$
   $$
   x = -\frac{2}{11}
   $$

5. **Substitute Back to Find the Other Variable**: Now substitute $$x$$ back into the first equation to find $$y$$:
   $$
   y = 2\left(-\frac{2}{11}\right) + 3
   $$
   $$
   y = -\frac{4}{11} + 3
   $$
   $$
   y = -\frac{4}{11} + \frac{33}{11} = \frac{29}{11}
   $$

6. **Final Solution**: The solution to the system of equations is:
   $$
   \left(x, y\right) = \left(-\frac{2}{11}, \frac{29}{11}\right)
   $$

This method is particularly effective when one equation is already solved for one variable, as it simplifies the process of finding the solution to the system.
The method of substitution works well when one equation has a variable isolated. By substituting that variable into the other equation, you can solve for the remaining variable and then find the solution to the system.

The method of substitution works particularly well if one or both of the equations has an isolated
variable on one side of the equation.

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