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A quadratic equation is graphed. Which of the following equations could be paired with the graphed equation to create a system of equations whose solution set is comprised of the points (2,-2) and (-3,3) ?

A quadratic equation is graphed. Which of the following equations could be paired with the graphed equation to create a system of equations whose solution set is comprised of the points (2,−2) (2,-2) and (−3,3) (-3,3) ?

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Q. A quadratic equation is graphed. Which of the following equations could be paired with the graphed equation to create a system of equations whose solution set is comprised of the points (2,−2) (2,-2) and (−3,3) (-3,3) ?
  1. Calculate Slope: We need to find a linear equation that passes through the points (2,−2)(2,-2) and (−3,3)(-3,3). To do this, we will use the point-slope form of a linear equation, which is y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope and (x1,y1)(x_1, y_1) is a point on the line.
  2. Use Point-Slope Form: First, we calculate the slope mm using the two given points (2,−2)(2,-2) and (−3,3)(-3,3). The slope formula is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. So, m=3−(−2)−3−2=5−5=−1m = \frac{3 - (-2)}{-3 - 2} = \frac{5}{-5} = -1.
  3. Simplify Equation: Now that we have the slope, we can use either of the two points to write the equation. Let's use the point (2,−2)(2,-2). Using the point-slope form, we get y−(−2)=−1(x−2)y - (-2) = -1(x - 2).
  4. Convert to Slope-Intercept Form: Simplify the equation to get it into slope-intercept form y=mx+by = mx + b.\newliney+2=−1(x−2)y + 2 = -1(x - 2)\newliney+2=−x+2y + 2 = -x + 2\newliney=−x+2−2y = -x + 2 - 2\newliney=−xy = -x
  5. Final Linear Equation: We have found the linear equation y=−xy = -x that passes through the points (2,−2)(2,-2) and (−3,3)(-3,3). This linear equation, when paired with the graphed quadratic equation, will create a system of equations whose solution set is comprised of the points (2,−2)(2,-2) and (−3,3)(-3,3).

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