Question was provided via image upload.

Question

Accepted Answer

### Problem 21
# Explanation:
## Step1: Substitute $ x = -6y + 7 $ into $ 2x + y = 3 $
$ 2(-6y + 7) + y = 3 $
## Step2: Simplify the equation
$ -12y + 14 + y = 3 $
$ -11y + 14 = 3 $
$ -11y = 3 - 14 $
$ -11y = -11 $
$ y = 1 $
## Step3: Substitute $ y = 1 $ back into $ x = -6y + 7 $
$ x = -6(1) + 7 $
$ x = 1 $
# Answer:
$ x = 1 $, $ y = 1 $

### Problem 22
# Explanation:
## Step1: Define variables
Let $ F $ be Frank’s present age, $ D $ be his daughter’s present age.
We know $ F + D = 74 $ (sum of present ages) and $ F + 3 = 3(D + 3) $ (in 3 years, Frank’s age = 3×daughter’s age).
## Step2: Simplify the second equation
$ F + 3 = 3D + 9 $
$ F = 3D + 6 $
## Step3: Substitute $ F = 3D + 6 $ into $ F + D = 74 $
$ 3D + 6 + D = 74 $
$ 4D + 6 = 74 $
$ 4D = 68 $
$ D = 17 $
# Answer:
Frank’s daughter’s age now is 17.

### Problem 23
# Explanation:
## Step1: Substitute $ y = 4x - 11 $ into $ -3x - 2y = -11 $
$ -3x - 2(4x - 11) = -11 $
## Step2: Simplify the equation
$ -3x - 8x + 22 = -11 $
$ -11x + 22 = -11 $
$ -11x = -33 $
$ x = 3 $
## Step3: Substitute $ x = 3 $ back into $ y = 4x - 11 $
$ y = 4(3) - 11 $
$ y = 12 - 11 $
$ y = 1 $
# Answer:
$ x = 3 $, $ y = 1 $

### Problem 24
# Explanation:
## Step1: Solve $ x + 4y = 2 $ for $ x $
$ x = 2 - 4y $
## Step2: Substitute $ x = 2 - 4y $ into $ 4x + 2y = -6 $
$ 4(2 - 4y) + 2y = -6 $
## Step3: Simplify the equation
$ 8 - 16y + 2y = -6 $
$ 8 - 14y = -6 $
$ -14y = -14 $
$ y = 1 $
## Step4: Substitute $ y = 1 $ back into $ x = 2 - 4y $
$ x = 2 - 4(1) $
$ x = -2 $
# Answer:
$ x = -2 $, $ y = 1 $

### Problem 25
# Explanation:
## Step1: Find slope of line 1 (points (25,50) and (20,40))
Slope $ m_1 = \frac{40 - 50}{20 - 25} = \frac{-10}{-5} = 2 $
## Step2: Find slope of line 2 (points (90,30) and (-20,-2))
Slope $ m_2 = \frac{-2 - 30}{-20 - 90} = \frac{-32}{-110} = \frac{16}{55} $
## Step3: Compare slopes
Since $ m_1
eq m_2 $, lines intersect at one point.
# Answer:
Number of solutions is 1.

### Problem 26
# Explanation:
## Step1: Convert $ \frac{x}{2} - \frac{y}{2} = 2 $ to slope - intercept form ($ y = mx + b $)
Multiply both sides by 2: $ x - y = 4 $
$ -y = -x + 4 $
$ y = x - 4 $ (slope $ m_1 = 1 $, y - intercept $ b_1 = -4 $)
## Step2: Convert $ 2x - y = 4 $ to slope - intercept form
$ -y = -2x + 4 $
$ y = 2x - 4 $ (slope $ m_2 = 2 $, y - intercept $ b_2 = -4 $)
## Step3: Compare slopes and y - intercepts
Since $ m_1
eq m_2 $, lines intersect at one point.
# Answer:
Number of solutions is 1.

### Problem 27
# Explanation:
## Step1: Convert $ -4x + 3y = 9 $ to slope - intercept form
$ 3y = 4x + 9 $
$ y = \frac{4}{3}x + 3 $ (slope $ m_1 = \frac{4}{3} $, y - intercept $ b_1 = 3 $)
## Step2: Convert $ -8x + 6y = 18 $ to slope - intercept form
$ 6y = 8x + 18 $
$ y = \frac{8}{6}x + 3 $
Simplify: $ y = \frac{4}{3}x + 3 $ (slope $ m_2 = \frac{4}{3} $, y - intercept $ b_2 = 3 $)
## Step3: Compare slopes and y - intercepts
Since $ m_1 = m_2 $ and $ b_1 = b_2 $, lines are coincident (infinite solutions).
# Answer:
Number of solutions is infinitely many.

### Problem 28
# Explanation:
The two equations are $ 4x + 2y = 10 $ and $ 4x + 2y = 10 $. They are the same equation, so every point on the line is a solution.
# Answer:
Number of solutions is infinitely many.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 21

Step1: Substitute \( x = -6y + 7 \) into \( 2x + y = 3 \)

Step2: Simplify the equation

Step3: Substitute \( y = 1 \) back into \( x = -6y + 7 \)

Step1: Define variables

Step2: Simplify the second equation

Step3: Substitute \( F = 3D + 6 \) into \( F + D = 74 \)

Step1: Substitute \( y = 4x - 11 \) into \( -3x - 2y = -11 \)

Step2: Simplify the equation

Step3: Substitute \( x = 3 \) back into \( y = 4x - 11 \)

Answer:

Problem 22