expand each expression by using the distributive property.
1. ( x^2y^3(3xy - 5y) )
2. ( -2x^3y^3(4x^2y - 3xy) )
evaluate each expression for the given values.
3. ( x^2y^3z ) if ( x = 3, y = -2 ), and ( z = 4 )
4. ( -x^2 - y^3 ) if ( x = -3 ) and ( y = -2 )
5. find the ( x )-intercept of the line ( 3x + 2y - 10 = 0 ).
6. identify the slope and ( y )-intercept of the line ( 2x - 5y - 6 = 0 ).
7. multiple choice what is the equation of the graphed line?
a ( y = -\frac{1}{3}x + 3 )
b ( y = -\frac{1}{3}x - 3 )
c ( y = -3x + 3 )
d ( y = -3x - 3 )
(graph: a coordinate plane with x-axis from -8 to 8 and y-axis from -8 to 8. a line passes through points, with y-intercept at -3 and x-intercept at -9? (approximate from graph))

Question

Accepted Answer

### Problem 1: Expand $ x^2y^3(3xy - 5y) $ using Distributive Property
# Explanation:
## Step 1: Apply Distributive Property ($ a(b - c)=ab - ac $)
$ x^2y^3 \cdot 3xy - x^2y^3 \cdot 5y $
## Step 2: Multiply coefficients and add exponents ( $ x^m \cdot x^n = x^{m + n} $, $ y^m \cdot y^n = y^{m + n} $)
For the first term: $ 3x^{2 + 1}y^{3 + 1}=3x^3y^4 $
For the second term: $ 5x^2y^{3 + 1}=5x^2y^4 $
So the expanded form is $ 3x^3y^4 - 5x^2y^4 $
# Answer: $ 3x^3y^4 - 5x^2y^4 $

### Problem 2: Expand $ -2x^3y^3(4x^2y - 3xy) $ using Distributive Property
# Explanation:
## Step 1: Apply Distributive Property ($ a(b - c)=ab - ac $)
$ -2x^3y^3 \cdot 4x^2y - (-2x^3y^3) \cdot 3xy $
## Step 2: Multiply coefficients and add exponents
First term: $ -8x^{3 + 2}y^{3 + 1}=-8x^5y^4 $
Second term: $ +6x^{3 + 1}y^{3 + 1}=6x^4y^4 $
So the expanded form is $ -8x^5y^4 + 6x^4y^4 $
# Answer: $ -8x^5y^4 + 6x^4y^4 $

### Problem 3: Evaluate $ x^2y^3z $ for $ x = 3 $, $ y = -2 $, $ z = 4 $
# Explanation:
## Step 1: Substitute the values into the expression
$ (3)^2(-2)^3(4) $
## Step 2: Calculate each power
$ 9 \cdot (-8) \cdot 4 $
## Step 3: Multiply the numbers
$ 9 \cdot (-32)= -288 $
# Answer: $ -288 $

### Problem 4: Evaluate $ -x^2 - y^3 $ for $ x = -3 $, $ y = -2 $
# Explanation:
## Step 1: Substitute the values into the expression
$ -(-3)^2 - (-2)^3 $
## Step 2: Calculate each power
$ -9 - (-8) $
## Step 3: Simplify the subtraction of a negative
$ -9 + 8 = -1 $
# Answer: $ -1 $

### Problem 5: Find the $ x $-intercept of $ 3x + 2y - 10 = 0 $
# Explanation:
## Step 1: Recall $ x $-intercept occurs at $ y = 0 $
Substitute $ y = 0 $ into the equation: $ 3x + 2(0) - 10 = 0 $
## Step 2: Solve for $ x $
$ 3x - 10 = 0 $
$ 3x = 10 $
$ x=\frac{10}{3} $
# Answer: $ \frac{10}{3} $ (or $ 3\frac{1}{3} $)

### Problem 6: Identify slope and $ y $-intercept of $ 2x - 5y - 6 = 0 $
# Explanation:
## Step 1: Rewrite in slope - intercept form ($ y = mx + b $, where $ m $ is slope, $ b $ is $ y $-intercept)
Solve for $ y $:
$ -5y=-2x + 6 $
$ y=\frac{-2x + 6}{-5}=\frac{2}{5}x-\frac{6}{5} $
## Step 2: Identify $ m $ and $ b $
Slope ($ m $) is $ \frac{2}{5} $, $ y $-intercept ($ b $) is $ -\frac{6}{5} $
# Answer: Slope is $ \frac{2}{5} $, $ y $-intercept is $ -\frac{6}{5} $

### Problem 7: Multiple Choice - Equation of the graphed line
# Explanation:
## Step 1: Recall slope - intercept form $ y = mx + b $, where $ b $ is $ y $-intercept (when $ x = 0 $)
From the graph, when $ x = 0 $, $ y=-3 $? Wait, no, looking at the graph: the line crosses the $ y $-axis at $ (0, - 3) $? Wait, no, the graph shows the line crosses the $ y $-axis at $ (0, - 3) $? Wait, no, let's check the points. The line passes through $ (- 3, - 2) $? Wait, no, let's use two points. The line passes through $ (- 3, - 2) $? Wait, the graph has a line with $ y $-intercept at $ (0, - 3) $? Wait, no, the options: Let's calculate slope. Take two points: from the graph, when $ x=-3 $, $ y = - 2 $? Wait, no, let's use the $ y $-intercept. The line crosses the $ y $-axis at $ (0, - 3) $? Wait, no, the options: Option B: $ y =-\frac{1}{3}x - 3 $, Option A: $ y =-\frac{1}{3}x + 3 $, Option C: $ y=-3x + 3 $, Option D: $ y=-3x - 3 $. Let's calculate slope. Let's take two points: from the graph, the line passes through $ (- 3, - 2) $ and $ (0, - 3) $? Wait, no, when $ x=-3 $, $ y=-2 $? Wait, no, let's use the formula for slope $ m=\frac{y_2 - y_1}{x_2 - x_1} $. Let's take two points: Let's say the line passes through $ (- 3, - 2) $ and $ (0, - 3) $. Then $ m=\frac{-3-(-2)}{0 - (-3)}=\frac{-1}{3}=-\frac{1}{3} $. And the $ y $-intercept ($ b $) is $ - 3 $ (when $ x = 0 $, $ y=-3 $). So the equation is $ y =-\frac{1}{3}x-3 $, which is Option B.
# Brief Explanations:
1. Find two points on the line (e.g., $ (-3, - 2) $ and $ (0, - 3) $).
2. Calculate slope $ m=\frac{-3 - (-2)}{0 - (-3)}=-\frac{1}{3} $.
3. $ y $-intercept $ b=-3 $ (from $ (0, - 3) $).
4. Equation is $ y =-\frac{1}{3}x-3 $, which matches Option B.
# Answer: B. $ y =-\frac{1}{3}x - 3 $

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QUESTION IMAGE

Question

Explanation:

Problem 1: Expand \( x^2y^3(3xy - 5y) \) using Distributive Property

Step 1: Apply Distributive Property (\( a(b - c)=ab - ac \))

Step 2: Multiply coefficients and add exponents ( \( x^m \cdot x^n = x^{m + n} \), \( y^m \cdot y^n = y^{m + n} \))

Step 1: Apply Distributive Property (\( a(b - c)=ab - ac \))

Step 2: Multiply coefficients and add exponents

Step 1: Substitute the values into the expression

Step 2: Calculate each power

Step 3: Multiply the numbers

Answer:

Problem 2: Expand \( -2x^3y^3(4x^2y - 3xy) \) using Distributive Property