[latex]\\begin{array}{l}\\text{ }2x+3y - 6z=1\\hfill \\\\ -4x - 6y+12z=-2\\hfill \\\\ \\text{ }x+2y+5z=10\\hfill \\end{array}[\/latex]<\/p>\n5.\u00a0Every system of equations can be solved graphically, by substitution, and by addition. However, systems of three equations become very complex to solve graphically so other methods are usually preferable.\n\n7. No\n\n9.\u00a0Yes\n\n11.\u00a0[latex]\\left(-1,4,2\\right)[\/latex]\n\n13.\u00a0[latex]\\left(-\\frac{85}{107},\\frac{312}{107},\\frac{191}{107}\\right)[\/latex]\n\n15.\u00a0[latex]\\left(1,\\frac{1}{2},0\\right)[\/latex]\n\n17.\u00a0[latex]\\left(4,-6,1\\right)[\/latex]\n\n19.\u00a0[latex]\\left(x,\\frac{1}{27}\\left(65 - 16x\\right),\\frac{x+28}{27}\\right)[\/latex]\n\n21.\u00a0[latex]\\left(-\\frac{45}{13},\\frac{17}{13},-2\\right)[\/latex]\n\n23.\u00a0No solutions exist\n\n25.\u00a0[latex]\\left(0,0,0\\right)[\/latex]\n\n27.\u00a0[latex]\\left(\\frac{4}{7},-\\frac{1}{7},-\\frac{3}{7}\\right)[\/latex]\n\n29.\u00a0[latex]\\left(7,20,16\\right)[\/latex]\n\n31.\u00a0[latex]\\left(-6,2,1\\right)[\/latex]\n\n33.\u00a0[latex]\\left(5,12,15\\right)[\/latex]\n\n35.\u00a0[latex]\\left(-5,-5,-5\\right)[\/latex]\n\n37.\u00a0[latex]\\left(10,10,10\\right)[\/latex]\n\n39.\u00a0[latex]\\left(\\frac{1}{2},\\frac{1}{5},\\frac{4}{5}\\right)[\/latex]\n\n41.\u00a0[latex]\\left(\\frac{1}{2},\\frac{2}{5},\\frac{4}{5}\\right)[\/latex]\n\n43.\u00a0[latex]\\left(2,0,0\\right)[\/latex]\n\n45.\u00a0[latex]\\left(1,1,1\\right)[\/latex]\n\n47.\u00a0[latex]\\left(\\frac{128}{557},\\frac{23}{557},\\frac{28}{557}\\right)[\/latex]\n\n49.\u00a0[latex]\\left(6,-1,0\\right)[\/latex]\n\n51.\u00a024, 36, 48\n\n53.\u00a070 grandparents, 140 parents, 190 children\n\n55.\u00a0Your share was $19.95, Sarah\u2019s share was $40, and your other roommate\u2019s share was $22.05.\n\n57.\u00a0There are infinitely many solutions; we need more information\n\n59.\u00a0500 students, 225 children, and 450 adults\n\n61.\u00a0The BMW was $49,636, the Jeep was $42,636, and the Toyota was $47,727.\n\n63.\u00a0$400,000 in the account that pays 3% interest, $500,000 in the account that pays 4% interest, and $100,000 in the account that pays 2% interest.\n\n65.\u00a0The United States consumed 26.3%, Japan 7.1%, and China 6.4% of the world\u2019s oil.\n\n67.\u00a0Saudi Arabia imported 16.8%, Canada imported 15.1%, and Mexico 15.0%\n\n69.\u00a0Birds were 19.3%, fish were 18.6%, and mammals were 17.1% of endangered species","rendered":"
1.\u00a0[latex]\\left(1,-1,1\\right)[\/latex]<\/p>\n
2.\u00a0No solution.<\/p>\n
3.\u00a0Infinite number of solutions of the form [latex]\\left(x,4x - 11,-5x+18\\right)[\/latex].<\/p>\n
1.\u00a0No, there can be only one, zero, or infinitely many solutions.<\/p>\n
3.\u00a0Not necessarily. There could be zero, one, or infinitely many solutions. For example, [latex]\\left(0,0,0\\right)[\/latex] is not a solution to the system below, but that does not mean that it has no solution.<\/p>\n
[latex]\\begin{array}{l}\\text{ }2x+3y - 6z=1\\hfill \\\\ -4x - 6y+12z=-2\\hfill \\\\ \\text{ }x+2y+5z=10\\hfill \\end{array}[\/latex]<\/p>\n
5.\u00a0Every system of equations can be solved graphically, by substitution, and by addition. However, systems of three equations become very complex to solve graphically so other methods are usually preferable.<\/p>\n
7. No<\/p>\n
9.\u00a0Yes<\/p>\n
11.\u00a0[latex]\\left(-1,4,2\\right)[\/latex]<\/p>\n
13.\u00a0[latex]\\left(-\\frac{85}{107},\\frac{312}{107},\\frac{191}{107}\\right)[\/latex]<\/p>\n
15.\u00a0[latex]\\left(1,\\frac{1}{2},0\\right)[\/latex]<\/p>\n
17.\u00a0[latex]\\left(4,-6,1\\right)[\/latex]<\/p>\n
19.\u00a0[latex]\\left(x,\\frac{1}{27}\\left(65 - 16x\\right),\\frac{x+28}{27}\\right)[\/latex]<\/p>\n
21.\u00a0[latex]\\left(-\\frac{45}{13},\\frac{17}{13},-2\\right)[\/latex]<\/p>\n
23.\u00a0No solutions exist<\/p>\n
25.\u00a0[latex]\\left(0,0,0\\right)[\/latex]<\/p>\n
27.\u00a0[latex]\\left(\\frac{4}{7},-\\frac{1}{7},-\\frac{3}{7}\\right)[\/latex]<\/p>\n
29.\u00a0[latex]\\left(7,20,16\\right)[\/latex]<\/p>\n
31.\u00a0[latex]\\left(-6,2,1\\right)[\/latex]<\/p>\n
33.\u00a0[latex]\\left(5,12,15\\right)[\/latex]<\/p>\n
35.\u00a0[latex]\\left(-5,-5,-5\\right)[\/latex]<\/p>\n
37.\u00a0[latex]\\left(10,10,10\\right)[\/latex]<\/p>\n
39.\u00a0[latex]\\left(\\frac{1}{2},\\frac{1}{5},\\frac{4}{5}\\right)[\/latex]<\/p>\n
41.\u00a0[latex]\\left(\\frac{1}{2},\\frac{2}{5},\\frac{4}{5}\\right)[\/latex]<\/p>\n
43.\u00a0[latex]\\left(2,0,0\\right)[\/latex]<\/p>\n
45.\u00a0[latex]\\left(1,1,1\\right)[\/latex]<\/p>\n
47.\u00a0[latex]\\left(\\frac{128}{557},\\frac{23}{557},\\frac{28}{557}\\right)[\/latex]<\/p>\n
49.\u00a0[latex]\\left(6,-1,0\\right)[\/latex]<\/p>\n
51.\u00a024, 36, 48<\/p>\n
53.\u00a070 grandparents, 140 parents, 190 children<\/p>\n
55.\u00a0Your share was $19.95, Sarah\u2019s share was $40, and your other roommate\u2019s share was $22.05.<\/p>\n
57.\u00a0There are infinitely many solutions; we need more information<\/p>\n
59.\u00a0500 students, 225 children, and 450 adults<\/p>\n
61.\u00a0The BMW was $49,636, the Jeep was $42,636, and the Toyota was $47,727.<\/p>\n
63.\u00a0$400,000 in the account that pays 3% interest, $500,000 in the account that pays 4% interest, and $100,000 in the account that pays 2% interest.<\/p>\n
65.\u00a0The United States consumed 26.3%, Japan 7.1%, and China 6.4% of the world\u2019s oil.<\/p>\n
67.\u00a0Saudi Arabia imported 16.8%, Canada imported 15.1%, and Mexico 15.0%<\/p>\n
69.\u00a0Birds were 19.3%, fish were 18.6%, and mammals were 17.1% of endangered species<\/p>\n\n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t