The line graphed on the grid represents the first of two equations in a system of linear equations.

If the graph of the second equation in the system passes through the points (−12,20)
and (4,12)
, which statement is true?
A.The only solution to the system is (0,14)
.
B.The system has an infinite number of solutions.
C.The system has no solution.
D.The only solution to the system is (10,5)
.

To determine the correct statement about the system of equations, we need to analyze the line formed by the two points given for the second equation: \((-12, 20)\) and \((4, 12)\).

1. **Calculate the slope (m)** of the line connecting these points:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 20}{4 - (-12)} = \frac{-8}{16} = -\frac{1}{2}
   \]

2. **Use the point-slope form** to write the equation of the line:
   \[
   y - y_1 = m(x - x_1)
   \]
   Using point \((4, 12)\):
   \[
   y - 12 = -\frac{1}{2}(x - 4)
   \]
   Simplifying:
   \[
   y - 12 = -\frac{1}{2}x + 2 \implies y = -\frac{1}{2}x + 14
   \]

3. **Examine the first line** (represented by the graph). We need to check its slope and y-intercept to see if the lines are parallel, coincident, or intersecting.

4. **Analysis**:
   - If the slopes are the same and the y-intercepts are different, the system has no solution (the lines are parallel).
   - If the slopes are different, there will be exactly one solution (the lines intersect).
   - If the slopes and y-intercepts are the same, there are infinite solutions (the lines are coincident).

**Conclusion**:
- Without the graph of the first equation, we cannot definitively determine the relationship between the two lines.
- However, if you check the slopes and the intercepts, you can conclude which statement is true.

From the options:
- A: (0, 14) could be a solution if it lies on both lines.
- B: Infinite solutions occur if both equations represent the same line.
- C: No solution if the lines are parallel.
- D: (10, 5) could also be a solution depending on the first line.

**To decide**: Check the first equation's slope and intercept against \(-\frac{1}{2}x + 14\). 

Do you have the equation for the first line, or can you provide more details about the graph?

The line graphed on the grid represents the first of two equations in a system of linear equations. If the graph of the second equation in the system passes through the points (−12,20) and (4,12), which statement is true?

Explore the analysis of a system of linear equations using the points (−12, 20) and (4, 12). This guide covers how to determine the nature of the solution based on the slopes and intercepts of the lines. Ideal for Grades 7-9 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation