Problem 61 - 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. $$ \left\\{\begin{array}{l} y=2 x+3 \\ 2 x-y=-3 \end{array}\right. $$

Step-by-Step Solution

Answer

Infinitely many solutions. The system is dependent and consistent.

Step 1: Write both equations in slope-intercept form

The first equation is already in slope-intercept form ( y = mx + b), which gives us y = 2x + 3.

Step 2: Rewrite the second equation in slope-intercept form

Start with 2x - y = -3. Solve for y by isolating it on one side: 2x - y = -3 y = 2x + 3.

Step 3: Compare the slopes and y-intercepts

The first equation is y = 2x + 3, and after rearranging, the second equation is also y = 2x + 3. Both equations have the same slope ( m = 2) and the same y-intercept ( b = 3).

Step 4: Determine the number of solutions

Since both equations are identical, they represent the same line and therefore have infinitely many solutions as every point on one line is also on the other.

Step 5: Classify the system of equations

This system is classified as dependent and consistent since both equations describe the same line.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial to solving linear systems efficiently. The slope-intercept form of a linear equation is given by: \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. The slope indicates the steepness of the line, showing how much \( y \) changes for a unit change in \( x \). For example, in the equation \( y = 2x + 3 \), the slope \( m \) is 2, and the y-intercept \( b \) is 3. This means the line crosses the y-axis at \( y = 3 \), and for every increase of 1 in \( x \), \( y \) increases by 2.

Consistent Systems

A consistent system of linear equations is a set of equations that has at least one solution. These systems can be solved by finding a common solution point (or points) that satisfies all of the equations in the system. For example, consider the following equations:
\ \[ y = 2x + 3 \] \ and \ \[ 2x - y = -3 \]. \
By converting both equations to slope-intercept form, we can compare their slopes and y-intercepts to determine consistency. Since these equations are identical after rearranging, every point on one line is also a point on the other line. Therefore, this system is consistent. Consistent systems can either have one solution (if the lines intersect at a single point) or infinitely many solutions (if the lines overlap exactly).

Dependent Systems

Dependent systems are a specific type of consistent systems where the equations describe the same line. In other words, all solutions of one equation are also solutions of the other equation. This results in infinitely many solutions, as every point along the line satisfies both equations.

For instance, in the given example: \ \[ y = 2x + 3 \] \ and \ \[ 2x - y = -3 \], \
both equations, once simplified, turn out to be the same line. They have identical slopes (2) and y-intercepts (3), indicating that they describe the same line. Dependent systems are classified as both consistent and dependent, highlighting the special case where there are infinite solutions.

Infinite Solutions

A system of linear equations has infinitely many solutions if the equations describe the same line. This means that every point on one line is also on the other line. These systems are both consistent and dependent. Infinite solutions occur because both equations are essentially the same.

Using the earlier example: \[ y = 2x + 3 \] and \[ 2x - y = -3 \], after rewriting the second equation in slope-intercept form, we get \( y = 2x + 3 \). Both equations depict the same line, so their solutions overlap entirely, leading to an infinite number of solutions. Identifying infinite solutions is straightforward: if the simplified forms of the equations are identical, the system has infinite solutions.