Question

use a graphing calculator to graph each system of equations. determine which quadrant the solution lies in
i
ii
iii
iv
3x + 2y = 10
5x - 2y = 6
x - y = -3
x + y = -7
y = 2x - 9
y = x - 5
5x - 2y = -10
3x + y = -7

Explanation:

Step1: Solve first system \(3x + 2y = 10\) and \(5x - 2y = 6\)

Add the two equations: \((3x + 2y)+(5x - 2y)=10 + 6\) → \(8x=16\) → \(x = 2\). Substitute \(x = 2\) into \(3x + 2y = 10\): \(3(2)+2y = 10\) → \(6 + 2y = 10\) → \(2y = 4\) → \(y = 2\). Solution \((2,2)\) is in Quadrant I.

Step2: Solve second system \(x - y = - 3\) and \(x + y = - 7\)

Add the two equations: \((x - y)+(x + y)=-3+(-7)\) → \(2x=-10\) → \(x=-5\). Substitute \(x = - 5\) into \(x - y = - 3\): \(-5 - y=-3\) → \(y=-2\). Solution \((-5,-2)\) is in Quadrant III.

Step3: Solve third system \(y = 2x - 9\) and \(y = x - 5\)

Set \(2x - 9=x - 5\) → \(2x-x=-5 + 9\) → \(x = 4\). Substitute \(x = 4\) into \(y = x - 5\): \(y=4 - 5=-1\). Solution \((4,-1)\) is in Quadrant IV.

Step4: Solve fourth system \(5x - 2y = - 10\) and \(3x + y = - 7\)

From \(3x + y=-7\), we get \(y=-3x - 7\). Substitute into \(5x - 2y=-10\): \(5x-2(-3x - 7)=-10\) → \(5x + 6x+14=-10\) → \(11x=-24\) → \(x=-\frac{24}{11}\approx - 2.18\). Then \(y=-3(-\frac{24}{11})-7=\frac{72}{11}-7=\frac{72 - 77}{11}=-\frac{5}{11}\approx - 0.45\). Solution \((-\frac{24}{11},-\frac{5}{11})\) is in Quadrant III.

Answer:

First system \((3x + 2y = 10, 5x - 2y = 6)\): Quadrant I
Second system \((x - y = - 3, x + y = - 7)\): Quadrant III
Third system \((y = 2x - 9, y = x - 5)\): Quadrant IV
Fourth system \((5x - 2y = - 10, 3x + y = - 7)\): Quadrant III