College Algebra CLEP: Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The solution to a system is the values of the variables that satisfy all equations simultaneously. These equations appear in real-world problems involving multiple constraints or conditions.

This lesson will cover methods to solve systems of equations: substitution, elimination, and special cases.

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1. Definition of a System of Linear Equations

A system of two linear equations in two variables can be written as:

where:

• $a_1,a_2,b_1,b_2,c_1,c_2$ are real numbers,
• $x$ and $y$ are variables.

A solution to the system is a pair $(x,y)$ that satisfies both equations.

Example:

Solve using elimination:

Step 1: Add the equations to eliminate $y$

Step 2: Simplify

Step 3: Substitute $x=3$ into the first equation

Solution:

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2. Solving Systems of Equations

There are three primary methods to solve a system of equations:

1. Substitution Method
2. Elimination Method
3. Graphical Interpretation (not covered in detail here)

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3. Substitution Method

The substitution method is useful when one equation is already solved for a variable or can be easily manipulated to do so.

Steps:

1. Solve one equation for one variable in terms of the other.
2. Substitute this expression into the second equation.
3. Solve for the remaining variable.
4. Use the found value to solve for the other variable.

Example:

Solve using substitution:

Step 1: Substitute $y=2x+1$ into the second equation

Step 2: Expand and solve for $x$

Step 3: Solve for $y$

Solution:

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4. Elimination Method

The elimination method is useful when both equations contain the variables in a similar format. It involves adding or subtracting equations to eliminate one variable.

Steps:

1. Multiply one or both equations if necessary to align coefficients.
2. Add or subtract the equations to eliminate one variable.
3. Solve for the remaining variable.
4. Substitute the found value into one of the original equations.

Example:

Solve using elimination:

Step 1: Add the equations to eliminate $y$

Step 2: Simplify

Step 3: Substitute $x=3$ into the first equation

Solution:

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5. Special Cases

Some systems have no solution or infinitely many solutions.

No Solution (Inconsistent System)

If elimination results in a false statement like:

the system has no solution (parallel lines).

Example:
$$
\begin{cases}
2x + 4y = 8 \
x + 2y = 5
\end{cases}
$$

Multiplying the second equation by 2:

Since $8\ne 10$, the system has no solution.

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Infinite Solutions (Dependent System)

If elimination results in a true statement like:

the system has infinitely many solutions (coincident lines).

Example:
$$
\begin{cases}
4x + 2y = 10 \
2x + y = 5
\end{cases}
$$

Multiplying the second equation by 2:

Since both equations are identical, the system has infinitely many solutions.

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6. Practice Problems

Solve the following systems of equations:

Try solving each system and check your answers!