Problem 47 - 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.) $$ \left\\{\begin{array}{l} x+y=0 \\ y=2 x-6 \end{array}\right. $$

Step-by-Step Solution

Answer

The lines intersect at the point (2, -2), which is the solution to the system.

Step 1: Rearrange the First Equation

The first equation of the system is given as \( x + y = 0 \). Rearranging this equation for \( y \), we get: \[ y = -x. \] This equation will be used to graph a line.

Step 2: Rearrange the Second Equation

The second equation of the system is \( y = 2x - 6 \). This equation is already solved for \( y \) and can be directly used to graph a line.

Step 3: Graph the Equations

Graph both equations on the same set of axes. - For \( y = -x \): This is a line passing through the origin with a slope of -1. - For \( y = 2x - 6 \): This line has a slope of 2 and y-intercept of -6 (crossing the y-axis at -6). Plot points for both lines and draw them on the graph.

Step 4: Identify the Intersection Point

The solutions to the system correspond to the points where the two lines intersect. By inspection of the graph, find this intersection point, which appears to be where the coordinates are fractions.

Step 5: Solve Algebraically to Find Exact Intersection

Set the equations for \( y \) equal to each other to find the exact intersection point algebraically: \[ -x = 2x - 6. \]Solve for \( x \):\[ -x - 2x = -6, \]\[ -3x = -6, \]\[ x = 2. \]Substitute back to find \( y \):\[ y = -x = -2. \]Thus, the intersection point is \( (2, -2) \).

Step 6: Verify the Solution

Verify that \( (2, -2) \) satisfies both original equations: - For \( x + y = 0 \): \( 2 + (-2) = 0 \) which is true.- For \( y = 2x - 6 \): \( -2 = 2(2) - 6 \), which simplifies to \( -2 = -2 \), also true. Therefore, the algebraic solution matches the point of intersection found graphically.

Key Concepts

Graphing Equations

Graphing equations involves plotting lines on a coordinate plane to visually find the solution of a system. For any linear equation, such as the ones in the exercise, it's crucial to identify two key elements:

• The slope, which shows how steep the line is.
• The y-intercept, which tells where the line crosses the y-axis.

For the first equation \(x + y = 0\), rearranging it gives \(y = -x\). This reveals a line with a slope of -1 passing through the origin.
Similarly, the second equation \(y = 2x - 6\) already shows a slope of 2 and a y-intercept of -6.
On graphing, these lines help easily visualize positions and intersections, guiding us to understand solutions where both conditions are met.

Intersection Point

The intersection point in a system of equations refers to the spot where two lines cross each other on a graph. This point represents the solution to the system as it satisfies both equations simultaneously. Visually, when we graph the equations \(y = -x\) and \(y = 2x - 6\), we look for the point where these two lines meet.
While the initial graphical inspection suggests the point might involve fractions, further algebraic methods provide precision. By setting \(-x = 2x - 6\), solving gives \(x = 2\). Substituting into either equation, we find \(y = -2\). Thus, the exact intersection point is \((2, -2)\).
It's crucial in graphing to verify that both equations yield the point to ensure accuracy.

Consistent and Independent Systems

Understanding if a system of equations is consistent and independent is key. A consistent system means there is at least one solution; an independent system suggests that the equations yield exactly one unique solution.
For the given system \(x + y = 0\) and \(y = 2x - 6\), once graphically represented, we identify that they intersect at a single point. This reveals the system is both consistent and independent.

• Their slopes are different: -1 for the first equation and 2 for the second.
• They do not overlap entirely nor are they parallel, ensuring a single clear solution.

Recognizing such traits ensures clarity in problem categorization, fundamental in algebra and graph interpretation.