Problem 67 - Algebra 1: Concepts and Skills

Question

Use the graphing method to tell how many solutions the system has. $$\begin{array}{r} {2 x-2 y=4} \\ {x+3 y=9} \end{array}$$

Step-by-Step Solution

Answer

The number of solutions is dependent on the graphical analysis. Observe the graph for intersections: single point represents one solution, overlapping lines imply infinite solutions and no intersection indicates no solution.

Step 1: Graph the first equation

The equation \(2x - 2y = 4\) can be rewritten into slope-intercept form \(y = mx + b\), where m represents the gradient and b the y-intercept. This results in \(y = x - 2\). Generate a set of points that satisfy this equation and plot on the graph.

Step 2: Graph the second equation

The equation \(x + 3y = 9\) is also transformed into slope-intercept form yielding \(y = \frac{1}{3}x - 3\). Identify points that fit this equation and mark them on the same graph.

Step 3: Analyze the graph for intersection points

The intersection point(s) of the two lines represent the solution(s) of the system. If these lines intersect at a single point, the system has one solution. If the lines overlap completely, it has infinite solutions. In case of no intersection, the system has no solution.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express linear equations that makes graphing easy. It is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

• The slope \(m\) tells us how steep the line is and the direction it goes. A positive slope means the line goes up while moving from left to right, and a negative slope means the line goes down.
• The y-intercept \(b\) is where the line crosses the y-axis. This point always has an x-coordinate of zero.

When you have an equation in standard form, like \(2x - 2y = 4\), you can rearrange it into slope-intercept form by solving for \(y\). For example, converting \(2x - 2y = 4\) to slope-intercept form results in \(y = x - 2\). This format is very useful for graphing, as it gives you clear starting and progressing points.

Intersection of Lines

When graphing systems of equations, the intersection of lines is a crucial concept. The intersection point where two lines on a graph meet represents the solution to the system of equations. This point satisfies both equations simultaneously. There are different types of possible intersections:

• Single intersection point: This indicates one unique solution for the system of equations. Both lines cross each other exactly once.
• No intersection: If the lines are parallel and do not cross, the system has no solutions. This generally happens when the lines have the same slope but different y-intercepts.
• Infinite intersections: If two lines lie on top of each other, they have infinitely many points of intersection, meaning the system has infinite solutions.

By analyzing the intersection of lines, we can conclude how many solutions a system of equations has.

Solving Linear Equations

Solving linear equations is a fundamental skill in finding solutions to systems of equations. These types of equations form straight lines when graphically represented. To solve linear equations:

• Rearrange: Get the equation into slope-intercept form \(y = mx + b\) if it's not already.
• Plot: Identify the y-intercept \(b\) and plot it on the graph first. Then use the slope \(m\) to determine the direction and steepness of the line, plotting more points along this direction.

When you have multiple linear equations, graph them on the same plane. The solution is found at the point where their graphs intersect. This graphical method is a visual way to check for solutions and understand the relationship between different linear equations.

Infinite Solutions

A system of equations with infinite solutions occurs when the equations describe the same line. When you graph such a system, both lines lie on top of each other completely. This situation arises when:

• Both equations have the same slope \(m\) and y-intercept \(b\). This means they are essentially the same line expressed differently.
• In the example problem, if transforming both original equations into slope-intercept form reveals identical lines, infinite solutions are present. This means every point on one line is also on the other.

Whenever you solve a system and find the same equation, it implies all possible \((x, y)\) points applicable to these equations are solutions. This concept is essential for understanding scenarios beyond just single points of intersection.