We suggest you use technology. Graph the region corresponding to th \[ \begin{array}{l} 2.3 x-2.4 y \leq 2.5 \\ 4.0 x-5.1 y \leq 4.3 \\ 6.1 x+6.7 y \leq 9.7 \end{array} \] Find the coordinates of all corners points (if any) to two decimal place \[ \begin{array}{l} (x, y)=(\square) \\ (x, y)=(\square) \end{array} \] Need Help? Read it
Solution
Step 1 ：Rewrite the inequalities as equations: \(2.3x - 2.4y = 2.5\), \(4.0x - 5.1y = 4.3\), and \(6.1x + 6.7y = 9.7\)
Step 2 ：Find the intersection of the first two lines. Multiply the first equation by 5.1 and the second equation by 2.4 to get: \(11.73x - 12.24y = 12.75\) and \(9.6x - 12.24y = 10.32\)
Step 3 ：Subtract the second equation from the first to get: \(2.13x = 2.43\). Solve for x to get: \(x = 1.14\)
Step 4 ：Substitute \(x = 1.14\) into the first equation to get: \(2.3(1.14) - 2.4y = 2.5\). Solve for y to get: \(y = 0.05\)
Step 5 ：\(\boxed{(1.14, 0.05)}\) is the intersection of the first two lines
Step 6 ：Find the intersection of the first and third lines. Multiply the first equation by 6.7 and the third equation by 2.4 to get: \(15.41x - 16.08y = 16.75\) and \(14.64x + 16.08y = 23.28\)
Step 7 ：Add the two equations to get: \(30.05x = 40.03\). Solve for x to get: \(x = 1.33\)
Step 8 ：Substitute \(x = 1.33\) into the first equation to get: \(2.3(1.33) - 2.4y = 2.5\). Solve for y to get: \(y = 0.23\)
Step 9 ：\(\boxed{(1.33, 0.23)}\) is the intersection of the first and third lines
Step 10 ：Find the intersection of the second and third lines. Multiply the second equation by 6.7 and the third equation by 5.1 to get: \(26.8x - 34.17y = 28.81\) and \(31.11x + 34.17y = 49.47\)
Step 11 ：Add the two equations to get: \(57.91x = 78.28\). Solve for x to get: \(x = 1.35\)
Step 12 ：Substitute \(x = 1.35\) into the second equation to get: \(4.0(1.35) - 5.1y = 4.3\). Solve for y to get: \(y = 0.22\)
Step 13 ：\(\boxed{(1.35, 0.22)}\) is the intersection of the second and third lines
Step 14 ：Therefore, the corner points of the region defined by the inequalities are \(\boxed{(1.14, 0.05)}\), \(\boxed{(1.33, 0.23)}\), and \(\boxed{(1.35, 0.22)}\)
