Problem 19 - Algebra and Trigonometry

Question

\(15-20\) m Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$ \left\\{\begin{array}{l}{-x+\frac{1}{2} y=-5} \\ {2 x-y=10}\end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions because both equations represent the same line.

Step 1: Convert Equations to Slope-Intercept Form

The goal is to convert both linear equations to the slope-intercept form, which is \(y = mx + b\).\ Start with the first equation \(-x + \frac{1}{2}y = -5\). First, add \(x\) to both sides: \[\frac{1}{2}y = x - 5\]Multiply every term by 2 to solve for \(y\):\[y = 2x - 10\] Now, let's rewrite the second equation: \(2x - y = 10\). Subtract \(2x\) from both sides:\[-y = -2x + 10\]Multiply every term by -1 to solve for \(y\):\[y = 2x - 10\]

Step 2: Compare the Forms of the Equations

Both equations are now written as \(y = 2x - 10\). This indicates that the two equations are identical, meaning each line has the same slope \(m = 2\) and y-intercept \(b = -10\).

Step 3: Graph the Equations

When graphed, both equations produce the same line. Plot the line with a slope of 2 (up 2 units, right 1 unit from any point on the line) and a y-intercept at \(b = -10\). Since the lines are identical, they will overlap completely.

Step 4: Determine the Number of Solutions

Since the lines overlap completely, every point on the line is a solution to both equations. Thus, there are infinitely many solutions to this system of equations.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to express the equation of a line using the formula \(y = mx + b\). This formula is very handy for graphing as it shows both the slope and the y-intercept:

• Slope (\(m\)): This represents how steep the line is. The slope is the change in the y-value for a one-unit change in the x-value.
• Y-intercept (\(b\)): This is the point where the line crosses the y-axis. It is the value of \(y\) when \(x = 0\).

To convert any linear equation into the slope-intercept form, just solve for \(y\). This often involves moving terms from one side of the equation to the other, simplifying, and sometimes multiplying or dividing all terms to isolate \(y\). This form makes it straightforward to identify how a line will behave on a graph.

Graphing Linear Equations

Graphing linear equations is an important skill that helps visualize how equations relate to each other in a coordinate plane. To graph a line using its equation in the slope-intercept form \(y = mx + b\):

• Start with the Y-Intercept: Begin plotting the line at the y-intercept (point \((0, b)\) on the y-axis).
• Use the Slope: From the initial point, use the slope \(m\) as a guide. For example, a slope of 2 means you move up 2 units for every 1 unit you move to the right.
• Draw the Line: Connect these points across the grid to reveal the line. Extending the line will confirm the linear relationship described by the equation.

By plotting the lines of a system of equations, you can easily see how they relate. Do they intersect, overlap, or run parallel? Each scenario has different implications for the system's solutions.

Infinite Solutions

A system of linear equations can have infinitely many solutions when the equations describe the same line. In simpler words, when graphed, the lines overlap completely throughout the coordinate plane.

• Identical Lines: Two or more equations that are identical provide every point on the line as a solution.
• Consistent and Dependent: These systems have infinite solutions since any point on the line satisfies both equations.

To confirm infinite solutions, you convert all equations to the slope-intercept form. If their slopes \(m\) and y-intercepts \(b\) are the same, the two lines indeed represent the same line. Recognizing when equations result in an infinite number of solutions is crucial for understanding the consistent overlap between equations.