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list (a) Determine two different points on the line \( 4 x+7 y+7=0 \). (b) Draw the graph of the line in part (a) in a coordinate system. (a) Determine two different points on the line \( 4 x+7 y+7=0 \). Choose the correct answer below. A. \( (0,-5),(7,3) \) B. \( (0,3),(7,-1) \) C. \( (0,-1),(7,3) \) D. \( (0,-1),(7,-5) \) stion 12

Ask by Rodriquez Klein. in the United States
Mar 14,2025

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The correct answer is D: \( (0, -1) \) and \( (7, -5) \).

Solución

We are given the line \[ 4x + 7y + 7 = 0. \] We want to find two different points that lie on the line. **Step 1. Choose a value for \( x \) and solve for \( y \).** Let \( x = 0 \). Substitute into the equation: \[ 4(0) + 7y + 7 = 0 \quad \Longrightarrow \quad 7y + 7 = 0. \] Solve for \( y \): \[ 7y = -7 \quad \Longrightarrow \quad y = -1. \] Thus, one point is \( (0, -1) \). **Step 2. Choose another value for \( x \) and solve for \( y \).** Let \( x = 7 \). Substitute into the equation: \[ 4(7) + 7y + 7 = 0 \quad \Longrightarrow \quad 28 + 7y + 7 = 0. \] Combine like terms: \[ 7y + 35 = 0. \] Solve for \( y \): \[ 7y = -35 \quad \Longrightarrow \quad y = -5. \] Thus, another point is \( (7, -5) \). **Step 3. Verify the options.** Option D lists the points \( (0, -1) \) and \( (7, -5) \). These match our calculated points. **Answer:** D

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To find points on the line \( 4x + 7y + 7 = 0 \), we can plug in values for \( x \) or \( y \) and solve for the other variable. Let's test option A: 1. For \( (0, -5) \): \[ 4(0) + 7(-5) + 7 = -35 + 7 = -28 \, (\text{not on the line}) \] 2. For \( (7, 3) \): \[ 4(7) + 7(3) + 7 = 28 + 21 + 7 = 56 \, (\text{not on the line}) \] Now let's check option C: 1. For \( (0, -1) \): \[ 4(0) + 7(-1) + 7 = -7 + 7 = 0 \, (\text{on the line}) \] 2. For \( (7, 3) \): \[ 4(7) + 7(3) + 7 = 28 + 21 + 7 = 56 \, (\text{not on the line}) \] Through this method you can see that neither of these options is yielding the right set of points, so let’s find two points on our own: If \( x = 0 \), \( 4(0) + 7y + 7 = 0 \Rightarrow 7y = -7 \Rightarrow y = -1 \) (Point is \( (0, -1) \)) If \( y = 0 \), \( 4x + 7(0) + 7 = 0 \Rightarrow 4x = -7 \Rightarrow x = -\frac{7}{4} \) (Point is \( (-\frac{7}{4}, 0) \)) So the correct points would be \( (0, -1) \) and \( (-\frac{7}{4}, 0) \). Now for the fun part – visualization! To graph the line \( 4x + 7y + 7 = 0 \): 1. Plot the points you found, like \( (0, -1) \) and \( (-\frac{7}{4}, 0) \). 2. Draw a straight line through these points extending infinitely; this line represents all possible solutions to the equation! Visual aids like graph paper or online graphing tools can make this step more engaging!

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