College Math: Algebra Linear-Systems - Solving Systems by Substitution Step-by-Step

Solving Systems by Substitution – Step-by-Step

What Is This?

Solving systems of linear equations by substitution is a fundamental technique in algebra for finding the solution to a system of two or more equations. This method involves expressing one variable in terms of the other and then substituting it into the other equation to solve for the remaining variable.

Why It Matters

Solving systems by substitution is a crucial skill in various fields, including physics, engineering, economics, and data analysis. For instance, in physics, it's used to find the trajectory of an object under the influence of multiple forces. In economics, it's applied to determine the optimal production levels of goods and services. In data analysis, it's used to identify the relationships between variables in a dataset.

Core Concepts

• Linear Equations: Equations of the form $ax + by = c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables.
• Substitution Method: A technique for solving systems of linear equations by expressing one variable in terms of the other and then substituting it into the other equation.
• Consistent System: A system of linear equations that has at least one solution.
• Inconsistent System: A system of linear equations that has no solution.

Step-by-Step: How to Approach Problems

Step 1: Identify the System of Equations

Read the problem carefully and identify the system of linear equations.

Step 2: Choose a Variable to Substitute

Select a variable from one equation and express it in terms of the other variable.

Step 3: Substitute the Expression into the Other Equation

Replace the chosen variable with its expression in the other equation.

Step 4: Solve for the Remaining Variable

Solve the resulting equation for the remaining variable.

Step 5: Check the Solution

Substitute the solution back into the original equations to verify that it satisfies both equations.

Solved Examples

Problem 1

Solve the system of equations:

$$\begin{aligned} 2x + 3y &= 7 \ x - 2y &= -3 \end{aligned}$$

Solution

Step 1: Identify the system of equations. The given system of equations is:

$$\begin{aligned} 2x + 3y &= 7 \ x - 2y &= -3 \end{aligned}$$

Step 2: Choose a variable to substitute. Let's choose the variable $x$ from the second equation and express it in terms of $y$.

$$x = -3 + 2y$$

Step 3: Substitute the expression into the other equation. Substitute the expression for $x$ into the first equation:

$$2(-3 + 2y) + 3y = 7$$

Step 4: Solve for the remaining variable. Expand and simplify the equation:

$$-6 + 4y + 3y = 7$$

Combine like terms:

$$7y = 13$$

Divide by 7:

$$y = \frac{13}{7}$$

Step 5: Check the solution. Substitute the value of $y$ back into the expression for $x$:

$$x = -3 + 2\left(\frac{13}{7}\right)$$

Simplify:

$$x = -3 + \frac{26}{7}$$

$$x = \frac{-21 + 26}{7}$$

$$x = \frac{5}{7}$$

The solution is $x = \frac{5}{7}$ and $y = \frac{13}{7}$.

Problem 2

Solve the system of equations:

$$\begin{aligned} x + 2y &= 4 \ 3x - 2y &= 5 \end{aligned}$$

Solution

Step 1: Identify the system of equations. The given system of equations is:

$$\begin{aligned} x + 2y &= 4 \ 3x - 2y &= 5 \end{aligned}$$

Step 2: Choose a variable to substitute. Let's choose the variable $x$ from the first equation and express it in terms of $y$.

$$x = 4 - 2y$$

Step 3: Substitute the expression into the other equation. Substitute the expression for $x$ into the second equation:

$$3(4 - 2y) - 2y = 5$$

Step 4: Solve for the remaining variable. Expand and simplify the equation:

$$12 - 6y - 2y = 5$$

Combine like terms:

$$-8y = -7$$

Divide by -8:

$$y = \frac{7}{8}$$

Step 5: Check the solution. Substitute the value of $y$ back into the expression for $x$:

$$x = 4 - 2\left(\frac{7}{8}\right)$$

Simplify:

$$x = 4 - \frac{7}{4}$$

$$x = \frac{16 - 7}{4}$$

$$x = \frac{9}{4}$$

The solution is $x = \frac{9}{4}$ and $y = \frac{7}{8}$.

Problem 3

Solve the system of equations:

$$\begin{aligned} x + y &= 2 \ 2x + 3y &= 7 \end{aligned}$$

Solution

Step 1: Identify the system of equations. The given system of equations is:

$$\begin{aligned} x + y &= 2 \ 2x + 3y &= 7 \end{aligned}$$

Step 2: Choose a variable to substitute. Let's choose the variable $x$ from the first equation and express it in terms of $y$.

$$x = 2 - y$$

Step 3: Substitute the expression into the other equation. Substitute the expression for $x$ into the second equation:

$$2(2 - y) + 3y = 7$$

Step 4: Solve for the remaining variable. Expand and simplify the equation:

$$4 - 2y + 3y = 7$$

Combine like terms:

$$y = 3$$

Step 5: Check the solution. Substitute the value of $y$ back into the expression for $x$:

$$x = 2 - 3$$

$$x = -1$$

The solution is $x = -1$ and $y = 3$.

Common Pitfalls & Mistakes

• Incorrect Substitution: Failing to substitute the expression for one variable into the other equation correctly.
• Insufficient Simplification: Failing to simplify the equation after substitution, leading to incorrect solutions.
• Ignoring Consistency: Failing to check the solution in both original equations, leading to incorrect or inconsistent solutions.

Best Practices & Study Tips

• Check Your Work: Always check the solution in both original equations to ensure consistency.
• Simplify Carefully: Simplify the equation after substitution to avoid errors.
• Use a Systematic Approach: Follow a systematic approach to solving the system, such as choosing a variable to substitute and expressing it in terms of the other variable.

Tools & Software

• Graphing Calculators: TI-84, Desmos
• Statistical Software: R, Python libraries like NumPy/SciPy, Excel
• Symbolic Math Tools: Wolfram Alpha, Symbolab

Real-World Use Cases

• Physics: Finding the trajectory of an object under the influence of multiple forces.
• Economics: Determining the optimal production levels of goods and services.
• Data Analysis: Identifying the relationships between variables in a dataset.

Check Your Understanding (MCQs)

Question 1

What is the first step in solving a system of linear equations by substitution?

A) Substitute the expression for one variable into the other equation. B) Choose a variable to substitute. C) Check the solution in both original equations. D) Simplify the equation after substitution.

Correct Answer: B) Choose a variable to substitute.

Explanation

The first step in solving a system of linear equations by substitution is to choose a variable to substitute. This variable should be expressed in terms of the other variable, which is then substituted into the other equation.

Question 2

What is the purpose of checking the solution in both original equations?

A) To ensure consistency between the two equations. B) To simplify the equation after substitution. C) To choose a variable to substitute. D) To express one variable in terms of the other variable.

Correct Answer: A) To ensure consistency between the two equations.

Explanation

Checking the solution in both original equations ensures that the solution satisfies both equations, which is a crucial step in solving systems of linear equations by substitution.

Question 3

What is the benefit of using a systematic approach to solving a system of linear equations by substitution?

A) It ensures consistency between the two equations. B) It simplifies the equation after substitution. C) It helps to avoid errors and ensure accurate solutions. D) It expresses one variable in terms of the other variable.

Correct Answer: C) It helps to avoid errors and ensure accurate solutions.

Explanation

Using a systematic approach to solving a system of linear equations by substitution helps to avoid errors and ensure accurate solutions by following a step-by-step process.