Question

How many solution(s) does this system of equations have? $$ \begin{array}{c} m+2 n=1 \\ 6 n+3 m=9 \end{array} $$ (A) None (B) 1 (C) 2 (D) 3

Short answer

The system has no solution; the lines are parallel.

Step-by-step solution

Identify the System

The system of equations given is \( m + 2n = 1 \) and \( 6n + 3m = 9 \). We need to determine how many solutions exist for this system of equations.

Rewrite Second Equation

Observe that the second equation is \( 6n + 3m = 9 \). We can simplify by dividing every term by 3, resulting in \( 2n + m = 3 \). This simplifies further to match the form \( m + 2n = 3 \).

Compare Equations

Now, compare the simplified second equation \( m + 2n = 3 \) with the first equation \( m + 2n = 1 \).

Analyze Result

The equations \( m + 2n = 1 \) and \( m + 2n = 3 \) represent parallel lines with the same slope but different intercepts. Such lines never intersect.

Key concepts

Linear Equations

Linear equations are the foundation of algebra, forming the simplest type of systematic mathematical relationships. In the context of systems of equations, a linear equation is an equation where each term is constant or the product of a constant and a single variable. For example, the equation \( m + 2n = 1 \) is a simple linear equation. Here, each term is either a constant or a constant times a variable.
The power of linear equations comes from their ability to model relationships where changes are consistent over time, such as speed or rate scenarios. They form straight lines when graphed on a Cartesian plane.

• The general form of a linear equation in two variables is \( ax + by = c \).
• Every solution to a linear equation corresponds to a point on its graph.

This principle is essential in solving systems of equations, as it allows us to graphically visualize and find potential intersections that represent solutions.

Equation Solutions

Finding solutions to systems of equations is all about determining where the equations intersect. A solution to a system of linear equations is a set of values for the variables that satisfy all the equations simultaneously.
For example, with the set of equations \( m + 2n = 1 \) and \( m + 2n = 3 \), you would be looking for values of \( m \) and \( n \) that make both equations true at the same time.

• Unique Solution: Occurs if two lines intersect at exactly one point.
• No Solution: Happens if the lines are parallel and distinct, meaning they never intersect.
• Infinitely Many Solutions: Occurs if the two lines are identical.

In this system, analyzing the simplified equations \( m + 2n = 1 \) and \( m + 2n = 3 \), you find that these lines are parallel, thus confirming that no solutions exist because the lines do not meet.

Parallel Lines

Parallel lines are straight lines in a plane that never meet, regardless of how far they are extended. When analyzing a system of equations, discovering parallel lines is crucial because it implies that the system has no solutions.
Parallel lines are characterized by having the same slope but different y-intercepts. In the case of our system of equations, both simplify to the form \( m + 2n \), with differing constant terms. This shows they share identical slopes.

• If two lines described by equations like \( y = mx + c_1 \) and \( y = mx + c_2 \) (where \( m \) is identical) have different intercepts (\( c_1 eq c_2 \)), they are parallel.
• The key to identifying parallel lines is checking whether the ratios of the coefficients of the variables are equal, while constant terms are not.

Understanding parallel lines in systems of equations helps in quickly determining when no solutions are possible. This ensures you don't expend unnecessary effort solving what is a non-intersecting system.