Problem 62 - Intermediate Algebra

Question

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) $$ \left\\{\begin{array}{l} 2 x-y=-4 \\ 2 y=4 x-6 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent; the lines are parallel and do not intersect.

Step 1: Solve for y in the First Equation

The system of equations requires each equation to be written in the form \( y = mx + b \). First, solve the equation \( 2x - y = -4 \) for \( y \). Add \( y \) to both sides to obtain \( 2x = y - 4 \). To isolate \( y \), add 4 to both sides and rewrite it as \( y = 2x + 4 \).

Step 2: Solve for y in the Second Equation

Proceed with the equation \( 2y = 4x - 6 \). Solve for \( y \) by dividing the entire equation by 2 to get \( y = 2x - 3 \). Both equations are now in the form \( y = mx + b \).

Step 3: Graph Both Equations

Graph the two equations \( y = 2x + 4 \) and \( y = 2x - 3 \). Recognize that both lines have the same slope \( m = 2 \) but different y-intercepts (4 and -3 respectively), indicating they are parallel and will never intersect.

Step 4: Determine the Relationship

Upon graphing the equations, observe that the lines are parallel (with the same slope but different y-intercepts). Parallel lines do not intersect, meaning there is no solution to the system of equations.

Key Concepts

Graphing Method

The graphing method is an intuitive way to solve a system of linear equations by visually identifying the solution on a graph. To apply this method, you need to rewrite each equation in slope-intercept form, which is expressed as \( y = mx + b \). Here:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, or the point where the line crosses the y-axis.

Once the equations are in this form, you plot each line on a graph. The coordinates where the lines intersect represent the solution to the system. In cases where the lines coincide, they intersect at every point on the line, suggesting infinitely many solutions. However, if they never intersect, it indicates there is no solution. Thus, the graphing method not only helps to find solutions but also visualize the relationship between multiple equations.

Parallel Lines

Parallel lines occur in a system of linear equations when both lines have the same slope but different y-intercepts. The slope indicates the steepness and direction of a line, and if two lines share this attribute but have different starting points along the y-axis, they will never cross. This results in lines that are always the same distance apart.When you graph a system and find that the lines are parallel, it confirms that the system is inconsistent. This means there's no common set of \( x \) and \( y \) values that will satisfy both equations simultaneously. Understanding the nature of parallel lines is crucial in recognizing when a system of equations will have no solutions.

No Solution

A system of linear equations has no solution when the lines represented by these equations don't intersect. In our scenario, the two given equations after converting to slope-intercept form were \( y = 2x + 4 \) and \( y = 2x - 3 \). Both lines have the identical slope of 2, signifying they run parallel to each other and hence, will never meet.In mathematical terms, a system of equations without a solution is called inconsistent. This means there's no combination of \( x \) and \( y \) values that will yield true statements in both equations simultaneously. A visual graph showing two parallel lines is an immediate indication that it is impossible to find a point that lies on both. No intersection means no solution, making the visual check an essential step in solving systems of equations.