Problem 54 - Elementary Algebra

Question

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations. $$ \left\\{\begin{array}{l} y=3 x+4 \\ 9 x-3 y=18 \end{array}\right. $$

Step-by-Step Solution

Answer

No solutions; the system is inconsistent.

Step 1: Convert Second Equation to Slope-Intercept Form

Start with the second equation: y = 3x + 4 Re-arrange the second equation to have it in the slope-intercept form, which is y = mx + b. Begin by isolating y in the equation 9x - 3y = 18.Subtract 9x from both sides: -3y = -9x + 18 Divide every term by -3: y = 3x - 6

Step 2: Compare the Two Equations

Compare the equations: 1) y = 3x + 4 2) y = 3x - 6 Both equations have the same slope (m = 3), but different y-intercepts (4 and -6).

Step 3: Determine the Number of Solutions and Classify the System

Since the slopes are the same and the y-intercepts are different, the two lines are parallel and will never intersect. Therefore, the system has no solutions. This makes it an inconsistent system.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a very common way to represent linear equations. It looks like this:
\( y = mx + b \)
where:

• \(y\) is the dependent variable)
• \(m\) is the slope (rise over run)
• \(x\) is the independent variable
• \(b\) is the y-intercept (where the line crosses the y-axis)

In our example, the first equation is already in slope-intercept form:
\( y = 3x + 4 \)
The slope (m) is 3, and the y-intercept (b) is 4.
The second equation, 9x - 3y = 18, needs rearranging to fit this form.
We divide by -3 to isolate y, and get:
\( y = 3x - 6 \)
The slope is 3, and the y-intercept is -6. Both equations have the same slope.

Parallel Lines

Parallel lines are fascinating in geometry. They run side by side and never touch, no matter how far they extend.
For lines to be parallel:

• They must have the same slope.
• Their y-intercepts must be different.

In our system of equations:
\( y = 3x + 4 \)
\( y = 3x - 6 \)
We see both lines have a slope (\(m\)) of 3, but different y-intercepts (4 and -6).
Since they share the same slope, they are parallel and they will never intersect.

Inconsistent System

When dealing with systems of linear equations, the terminology can sometimes be tricky. An inconsistent system is one where there are no solutions.
This occurs because:

• The lines are parallel.
• They never meet.

In our case,
\( y = 3x + 4 \)
\( y = 3x - 6 \)
Since the lines are parallel, there’s no point (x, y) that satisfies both equations at the same time. Therefore, our system is classified as inconsistent.
Understanding this makes it easier to determine the nature of the solutions without graphing the lines.