Infinite Solutions

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Infinite Solutions · Grade 8 · Worksheet 3

1. A line is drawn on a coordinate plane that passes through the points (8, 14) and (12, 20). Another line is drawn that passes through the points (10, 17) and (14, 23). These two lines are graphed on the same coordinate plane. How many solutions (intersection points) do the equations representing these two lines have? Answer: infinitely many Solution: Find the slope of the first line using the points (8, 14) and (12, 20). Slope = (20 - 14) / (12 - 8) = 6 / 4 = 3/2. Find the slope of the second line using the points (10, 17) and (14, 23).
   Full step-by-step solution

   Step 1: Find the slope of the first line using the points (8, 14) and (12, 20). Slope = (20 - 14) / (12 - 8) = 6 / 4 = 3/2. Step 2: Find the slope of the second line using the points (10, 17) and (14, 23). Slope = (23 - 17) / (14 - 10) = 6 / 4 = 3/2. Step 3: Both lines have the same slope (3/2), so they are either parallel or the same line. Step 4: Check if they are the same line by seeing if (8, 14) lies on the second line. Use point-slope form for the second line: y - 17 = (3/2)(x - 10). Substitute x = 8: y - 17 = (3/2)(8 - 10) = (3/2)(-2) = -3. So y = 17 - 3 = 14. The point (8, 14) satisfies the equation of the second line. Step 5: Since the lines have the same slope and share at least one point, they are actually the same line. Therefore, they intersect at every point, giving infinitely many solutions. The answer is infinitely many.

2. A triangle is drawn on a coordinate plane with vertices at A(1, 2), B(5, 2), and C(3, 6). A line is drawn from point A to the midpoint of side BC. What are the coordinates of this midpoint? Answer: (4, 4) Solution: Identify the endpoints of side BC: B(5, 2) and C(3, 6). Calculate the x-coordinate of the midpoint: (5 + 3) ÷ 2 = 8 ÷ 2 = 4. Calculate the y-coordinate of the midpoint: (2 + 6) ÷ 2 = 8 ÷ 2 = 4.
   Full step-by-step solution

   Step 1: Identify the endpoints of side BC: B(5, 2) and C(3, 6). Step 2: Calculate the x-coordinate of the midpoint: (5 + 3) ÷ 2 = 8 ÷ 2 = 4. Step 3: Calculate the y-coordinate of the midpoint: (2 + 6) ÷ 2 = 8 ÷ 2 = 4. Step 4: Combine the coordinates: (4, 4). The midpoint of side BC is (4, 4).

3. 4(3x - 2) + 8 = 12x = ? Answer: 0 Solution: Distribute the 4 on the left side: 4(3x - 2) = 12x - 8 The equation becomes: 12x - 8 + 8 = 12x Combine like terms: -8 + 8 = 0, so we have 12x = 12x Subtract 12x from both sides: 12x - 12x = 12x - 12x This gives us 0 = 0 Since this is always true regardless of x, the equation has infinitely many…
   Full step-by-step solution

   Step 1: Distribute the 4 on the left side: 4(3x - 2) = 12x - 8 Step 2: The equation becomes: 12x - 8 + 8 = 12x Step 3: Combine like terms: -8 + 8 = 0, so we have 12x = 12x Step 4: Subtract 12x from both sides: 12x - 12x = 12x - 12x Step 5: This gives us 0 = 0 Step 6: Since this is always true regardless of x, the equation has infinitely many solutions. The question asks for the value of the expression 12x = ?, but since the equation is an identity, we need to find what makes it true for all x. Looking at the original equation 4(3x - 2) + 8 = 12x, when we simplify we get 12x = 12x, which means both sides are equal for any value of x. The question is asking for the value of the expression 12x = ?, but in the context of the equation being an identity, the answer is 0 because when we rearrange we get 0 = 0. The answer is 0.

4. Liam is designing a rectangular garden for his school's science project. The length of the garden is 3 meters more than twice its width. The perimeter of the garden is 36 meters. What are the dimensions of Liam's garden? Answer: width = 5 meters, length = 13 meters Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). \( P = 2 \times (\text{length} + \text{width}) \).
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). --- **Step 2: Write the perimeter formula** The perimeter \( P \) of a rectangle is: \( P = 2 \times (\text{length} + \text{width}) \). We are told \( P = 36 \), so: \( 2 \times (l + w) = 36 \). --- **Step 3: Substitute the expression for length** Since \( l = 2w + 3 \), \( 2 \times ( (2w + 3) + w ) = 36 \). --- **Step 4: Simplify inside the parentheses** \( (2w + 3) + w = 3w + 3 \). So: \( 2 \times (3w + 3) = 36 \). --- **Step 5: Solve for \( w \)** Divide both sides by 2: \( 3w + 3 = 18 \). Subtract 3 from both sides: \( 3w = 15 \). Divide by 3: \( w = 5 \). --- **Step 6: Find length** \( l = 2w + 3 = 2 \times 5 + 3 = 10 + 3 = 13 \). --- **Step 7: Final answer** Width = 5 meters, Length = 13 meters.

5. A rectangular garden is drawn on a coordinate plane with corners at (2, 1), (8, 1), (8, 5), and (2, 5). A diagonal line is drawn from the bottom-left corner (2, 1) to the top-right corner (8, 5). Find the equation of this diagonal line in slope-intercept form (y = mx + b). Answer: y = (2/3)x - 1/3 Solution: Find the slope (m) of the line. The slope formula is m = (y2 - y1) / (x2 - x1). Let (x1, y1) = (2, 1) and (x2, y2) = (8, 5).
   Full step-by-step solution

   Let's find the equation of the diagonal line from (2, 1) to (8, 5) in slope-intercept form y = mx + b. Step 1: Find the slope (m) of the line. The slope formula is m = (y2 - y1) / (x2 - x1). Let (x1, y1) = (2, 1) and (x2, y2) = (8, 5). So m = (5 - 1) / (8 - 2) = 4 / 6 = 2/3. The slope is 2/3. Step 2: Use the slope and one point to find the y-intercept (b). We can use the point-slope form: y - y1 = m(x - x1). Using (x1, y1) = (2, 1) and m = 2/3: y - 1 = (2/3)(x - 2) Step 3: Solve for y to put into slope-intercept form. y - 1 = (2/3)x - (2/3)*2 y - 1 = (2/3)x - 4/3 Now add 1 to both sides: y = (2/3)x - 4/3 + 1 Note that 1 = 3/3, so: y = (2/3)x - 4/3 + 3/3 y = (2/3)x - 1/3 Step 4: Verify with the other point (8, 5). Substitute x = 8 into our equation: y = (2/3)*8 - 1/3 = 16/3 - 1/3 = 15/3 = 5. This matches the y-coordinate of the point (8, 5), so the equation is correct. Final answer: y = (2/3)x - 1/3

6. A rectangular prism is drawn on a coordinate plane with vertices at (0,0,0), (6,0,0), (6,4,0), (0,4,0), (0,0,3), (6,0,3), (6,4,3), and (0,4,3). A diagonal is drawn from the vertex (0,0,0) to the opposite vertex (6,4,3). What is the length of this space diagonal? Round your answer to the nearest hundredth. Answer: 7.81 Solution: Identify the coordinates of the two vertices: (0,0,0) and (6,4,3) Calculate the differences in each coordinate: x-difference = 6-0 = 6, y-difference = 4-0 = 4, z-difference = 3-0 = 3 Apply the 3D distance formula: distance = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2) Substitute the values: distance…
   Full step-by-step solution

   Step 1: Identify the coordinates of the two vertices: (0,0,0) and (6,4,3) Step 2: Calculate the differences in each coordinate: x-difference = 6-0 = 6, y-difference = 4-0 = 4, z-difference = 3-0 = 3 Step 3: Apply the 3D distance formula: distance = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2) Step 4: Substitute the values: distance = sqrt(6^2 + 4^2 + 3^2) = sqrt(36 + 16 + 9) = sqrt(61) Step 5: Calculate sqrt(61) ≈ 7.8102 Step 6: Round to the nearest hundredth: 7.81 The answer is 7.81.

7. Sophia is helping her grandmother organize a craft fair. They are selling two types of handmade soaps: lavender and oatmeal. The cost to make each lavender soap is $9, and the cost to make each oatmeal soap is $7. They want to sell the soaps in bundles. For one bundle, they use 3 lavender soaps and 2 oatmeal soaps. The total cost to make the soaps for this bundle is $41. However, Sophia notices that if they instead use 6 lavender soaps and 4 oatmeal soaps for a larger bundle, the total cost would be $82. How many solutions exist for the cost of a single lavender soap and a single oatmeal soap that satisfy both bundle conditions? Answer: Infinitely many solutions Solution: Let x be the cost of one lavender soap and y be the cost of one oatmeal soap. From the first bundle: 3x + 2y = 41 From the second bundle: 6x + 4y = 82 Notice that the second equation is exactly 2 times the first equation: 2(3x + 2y) = 2(41) gives 6x + 4y = 82.
   Full step-by-step solution

   Step 1: Let x be the cost of one lavender soap and y be the cost of one oatmeal soap. Step 2: From the first bundle: 3x + 2y = 41 Step 3: From the second bundle: 6x + 4y = 82 Step 4: Notice that the second equation is exactly 2 times the first equation: 2(3x + 2y) = 2(41) gives 6x + 4y = 82. Step 5: Since both equations represent the same line, any point (x, y) that satisfies one equation will satisfy the other. Step 6: For example, if x = 9, then 3(9) + 2y = 41 gives 27 + 2y = 41, so 2y = 14 and y = 7. The point (9, 7) works. If x = 5, then 3(5) + 2y = 41 gives 15 + 2y = 41, so 2y = 26 and y = 13. The point (5, 13) also works. Step 7: Since there are infinitely many pairs (x, y) that satisfy both equations, the system has infinitely many solutions.