Math Problem Statement

The graph of linear equation A passes through the points ( − 7 ,

4 ) and ( 3 , − 10 ) , while the graph of linear equation B passes through the points ( − 7 ,

4 ) and ( 5 ,

11 ) . Which of these is a solution to the system of equations consisting of linear equation A and linear equation B? A. ( − 7 ,

4 )

B. ( 3 , − 10 )

C. ( 5 ,

11 )

D. ( 7 ,

4 )

Solution

We are given two linear equations, A and B, that pass through the following points:

  • Linear equation A: passes through points (7,4)(-7, 4) and (3,10)(3, -10)
  • Linear equation B: passes through points (7,4)(-7, 4) and (5,11)(5, 11)

To find which point is a solution to the system of equations, we need to identify a point that is common to both lines. That is, the point must lie on both lines.

Step 1: Identify the common point between the two lines

The point (7,4)(-7, 4) appears in both sets of points for the two equations:

  • Linear equation A passes through (7,4)(-7, 4) and (3,10)(3, -10).
  • Linear equation B passes through (7,4)(-7, 4) and (5,11)(5, 11).

Thus, the point (7,4)(-7, 4) is the only point common to both lines.

Step 2: Verify the options

  • Option A: (7,4)(-7, 4) is the common point for both equations.
  • Option B: (3, -10) only lies on equation A, not equation B.
  • Option C: (5, 11) only lies on equation B, not equation A.
  • Option D: (7, 4) does not lie on either line.

Conclusion:

The solution to the system of equations is Option A: (7,4)(-7, 4).

Would you like more details on how to derive the equations of these lines or any other clarification?

Here are some related questions you might find interesting:

  1. How do you find the equation of a line given two points?
  2. What is the slope-intercept form of a linear equation?
  3. How do you determine if two lines are parallel or perpendicular?
  4. How can you solve a system of linear equations graphically?
  5. What does the point of intersection represent in a system of equations?

Tip: When dealing with systems of linear equations, if two lines intersect at a single point, that point is the solution to the system.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Graphing Linear Equations

Formulas

Point-Slope Form of Linear Equation

Theorems

Intersection of Linear Equations

Suitable Grade Level

Grades 8-10