Which point would be a solution to the system of linear inequalities shown below?
y, is greater than, minus, 4, x, plus, 6, y, is greater than, one third, x, minus, 7
y>−4x+6y> 
3
1
​
 x−7

We are given a system of two linear inequalities:

1. \( y > -4x + 6 \)
2. \( y > \frac{1}{3}x - 7 \)

To find a solution, we need to determine which point satisfies both inequalities. Let's break this down:

### 1st inequality: \( y > -4x + 6 \)
This represents a line with a slope of \(-4\) and a y-intercept of \(6\). Since we are looking for points where \( y \) is greater than \( -4x + 6 \), any point above this line will satisfy this inequality.

### 2nd inequality: \( y > \frac{1}{3}x - 7 \)
This represents a line with a slope of \(\frac{1}{3}\) and a y-intercept of \(-7\). For this inequality, we are looking for points where \( y \) is greater than \( \frac{1}{3}x - 7 \), meaning points above this line will satisfy the second inequality.

#### To determine which point satisfies both inequalities, let's test a few points:

1. **(0, 0)**:
   - For \( y > -4x + 6 \): \( 0 > -4(0) + 6 \), or \( 0 > 6 \) → **False**.
   - For \( y > \frac{1}{3}x - 7 \): \( 0 > \frac{1}{3}(0) - 7 \), or \( 0 > -7 \) → **True**.
   - Since the first inequality is false, **(0, 0) is not a solution**.

2. **(2, 0)**:
   - For \( y > -4x + 6 \): \( 0 > -4(2) + 6 \), or \( 0 > -8 + 6 \), or \( 0 > -2 \) → **True**.
   - For \( y > \frac{1}{3}x - 7 \): \( 0 > \frac{1}{3}(2) - 7 \), or \( 0 > \frac{2}{3} - 7 \), or \( 0 > -6\frac{1}{3} \) → **True**.
   - Since both inequalities are true, **(2, 0) is a solution**.

Thus, the point \( (2, 0) \) satisfies both inequalities.

Would you like more details or explanations about this?

### Related Questions:
1. How do you graph systems of linear inequalities?
2. What does it mean geometrically when two inequalities form a solution region?
3. How do slope and y-intercept affect the graph of a linear inequality?
4. What is the importance of testing points in solving inequalities?
5. Can linear inequalities ever share the same boundary line?

**Tip**: When solving inequalities graphically, always check points both above and below the line to see which side satisfies the inequality!

Which point would be a solution to the system of linear inequalities shown below? y > -4x + 6, y > 1/3x - 7

Discover how to find a solution to the system of linear inequalities y > -4x + 6 and y > 1/3x - 7. This detailed solution involves graphing the inequalities and testing points to determine which region satisfies both inequalities. Suitable for students in Grades 9-10.

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