Solving Systems By Graphing Quiz 1 Look At The Graph Below ✓ Solved

Solving Systems By Graphing Quiz1 Look At The Graph Below How Many

Analyze the given system of linear equations to determine the number of solutions based on their graphical representation. Typically, systems of linear equations can have no solution, exactly one solution, infinitely many solutions, or in some cases, are impossible to determine without more information.

Question 1 asks: How many solutions does the system of equations have? The options are:

  • a. No Solution
  • b. One Solution
  • c. Two Solutions
  • d. Infinite Solutions

Question 2 repeats a similar query: How many solutions does the given system of equations have? Same options apply.

The next question refers to the graphs of y = x + 4 and y = -2x + 1. By analyzing their intersection point, the solution to the system can be identified from these options:

  • a. (-4, 0)
  • b. (-1, 3)
  • c. (0, 1)
  • d. (0, ?)

Another problem explores the nature of the graph when the solution is the point (2, -3). If a system has exactly this solution, the graph involves features such as:

  • a. Parallel lines
  • b. Lines that intersect at (2, -3)
  • c. Identical lines
  • d. A y-intercept of ?

Subsequently, the question asks: What is the number of solutions for the system of equations? The potential answers are:

  • a. One solution
  • b. No solutions
  • c. Infinite solutions
  • d. Impossible to determine

For the system y = -3x + 2 and y = 4x + 2, both lines share the same y-intercept at (0, 2). The question is: Which statement accurately describes the system? Options are:

  • a. There are no solutions to the system
  • b. There is at least one solution to the system

The problem also asks for the x-coordinate of the intersection point of y = -2x + 11 and y = x - 4. The options are:

  • a. 5
  • b. 1
  • c. -1
  • d. -

Finally, it questions whether a given ordered pair is a solution to the system, prompting the answer: True or False.

Sample Paper For Above instruction

Graphing systems of linear equations is a fundamental method in algebra to visualize the solution set and understand the relationships between different lines. The graphical approach involves plotting each equation as a line on the coordinate plane and analyzing their intersections. The nature of these intersections indicates the number of solutions the system has, which can be zero, one, or infinitely many.

When two lines are parallel, they never intersect, indicating no solutions to the system—this corresponds to option (a) "No Solution." Conversely, if the lines intersect at exactly one point, the system has one unique solution, corresponding to option (b) "One Solution." If the lines are coincident, meaning they are overlapping or exactly the same line, then the system has infinitely many solutions; every point on the line is a solution, which is option (d) "Infinite Solutions."

Analyzing the specific equations y = x + 4 and y = -2x + 1, their intersection point can be found algebraically or graphically. To find the solution, set the equations equal: x + 4 = -2x + 1. Solving for x gives: x + 2x = 1 - 4, which simplifies to 3x = -3, so x = -1. Substituting back into y = x + 4 yields y = -1 + 4 = 3. Thus, the intersection point is (-1, 3), aligning with option (b).

This point confirms that the solution to the system y = x + 4 and y = -2x + 1 is (-1, 3). Visualization of the graphs would show two lines crossing at this point, indicating a single, unique solution.

Regarding the scenario where the system's solution is (2, -3), the two lines would intersect at this exact coordinate. If the graph confirms that the two lines cross at (2, -3), then the system's solutions are not parallel or coincident but intersecting at that point. The nature of the lines—whether they are parallel, overlapping, or intersecting—determines the number of solutions.

In the case of y = -3x + 2 and y = 4x + 2, both share the same y-intercept at (0, 2). Since they have different slopes (-3 and 4), they are not parallel and will intersect at exactly one point. Setting -3x + 2 = 4x + 2 and solving, we find: -3x + 2 = 4x + 2, which simplifies to -3x = 4x, or -3x - 4x = 0, resulting in -7x = 0, so x = 0. Substituting x = 0 into either equation gives y = 2, confirming the intersection point (0, 2). Therefore, the system has a single solution, aligned with option (b).

To find the intersection point of y = -2x + 11 and y = x - 4, set the equations equal: -2x + 11 = x - 4. Simplify: -2x - x = -4 - 11, which gives -3x = -15. Solving for x: x = 5. This shows that the x-coordinate of the intersection is 5, consistent with option (a).

Determining whether a specific ordered pair is a solution involves substituting the x and y values into both equations to see if they satisfy both. For example, if the pair (x, y) is (2, -3), substitute into both equations: for y = -3x + 2, plugging x=2 gives y = -3(2) + 2 = -6 + 2 = -4, which does not equal -3; hence, this pair is not a true solution. Therefore, the answer to whether (2, -3) is a solution is False.

References

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