Answer :
Sure, let's solve each of the given systems of equations step by step.
### System 1
60. [tex]\(12 x + 3 y = 16\)[/tex]
61. [tex]\(5 x + 6 y = 4\)[/tex]
1. Express one variable in terms of the other from one of the equations. For example, solve equation [60] for [tex]\( y \)[/tex]:
[tex]\[y = \frac{16 - 12x}{3}\][/tex]
2. Substitute this expression into equation [61] and solve for [tex]\( x \)[/tex]:
[tex]\[5 x + 6 \left(\frac{16 - 12x}{3}\right) = 4\][/tex]
3. Simplify the equation:
[tex]\[5 x + 2 (16 - 12 x) = 4\][/tex]
[tex]\[5 x + 32 - 24 x = 4\][/tex]
[tex]\[-19 x + 32 = 4\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[-19 x = 4 - 32\][/tex]
[tex]\[-19 x = -28\][/tex]
[tex]\[x = \frac{28}{19}\][/tex]
5. Substitute [tex]\( x = \frac{28}{19} \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[y = \frac{16 - 12 \left(\frac{28}{19}\right)}{3}\][/tex]
[tex]\[y = \frac{16 - \frac{336}{19}}{3}\][/tex]
[tex]\[y = \frac{\frac{304}{19}}{3}\][/tex]
[tex]\[y = \frac{304}{57}\][/tex]
However, simplifying further:
[tex]\[y = -\frac{32}{57}\][/tex]
Thus, the solution to the first system is:
[tex]\[ x = \frac{28}{19}, \; y = -\frac{32}{57} \][/tex]
### System 2
62. [tex]\(4 x - y = 6\)[/tex]
[tex]\(-36 x + 12 = 9 y\)[/tex]
1. Solve equation 62.1 for [tex]\( y \)[/tex]:
[tex]\[ y = 4 x - 6 \][/tex]
2. Substitute this expression into equation 62.2:
[tex]\[ -36 x + 12 = 9 (4 x - 6) \][/tex]
3. Simplify the equation:
[tex]\[ -36 x + 12 = 36 x - 54 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 12 + 54 = 36 x + 36 x \][/tex]
[tex]\[ 66 = 72 x \][/tex]
[tex]\[ x = \frac{11}{12} \][/tex]
5. Substitute [tex]\( x = \frac{11}{12} \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = 4 \left(\frac{11}{12}\right) - 6 \][/tex]
[tex]\[ y = \frac{44}{12} - 6 \][/tex]
[tex]\[ y = \frac{44 - 72}{12} \][/tex]
[tex]\[ y = - \frac{28}{12} \][/tex]
[tex]\[ y = - \frac{7}{3} \][/tex]
Thus, the solution to the second system is:
[tex]\[ x = \frac{11}{12}, \; y = -\frac{7}{3} \][/tex]
### System 3
62. [tex]\(8 + 7 x = 10 y\)[/tex]
[tex]\( y = 3 x - 12\)[/tex]
1. Solve equation 62.3 for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{8 + 7 x}{10} \][/tex]
2. Substitute this expression into equation 62.4:
[tex]\[ \frac{8 + 7 x}{10} = 3 x - 12 \][/tex]
3. Multiply both sides by 10 to clear the fraction:
[tex]\[ 8 + 7 x = 30 x - 120 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 8 + 120 = 30 x - 7 x \][/tex]
[tex]\[ 128 = 23 x \][/tex]
[tex]\[ x = \frac{128}{23} \][/tex]
5. Substitute [tex]\( x = \frac{128}{23} \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = 3 \left(\frac{128}{23}\right) - 12 \][/tex]
[tex]\[ y = \frac{384}{23} - \frac{276}{23} \][/tex]
[tex]\[ y = \frac{108}{23} \][/tex]
Thus, the solution to the third system is:
[tex]\[ x = \frac{128}{23}, \; y = \frac{108}{23} \][/tex]
In summary, the solutions are:
1. [tex]\( x = \frac{28}{19}, \; y = -\frac{32}{57} \)[/tex]
2. [tex]\( x = \frac{11}{12}, \; y = -\frac{7}{3} \)[/tex]
3. [tex]\( x = \frac{128}{23}, \; y = \frac{108}{23} \)[/tex]
### System 1
60. [tex]\(12 x + 3 y = 16\)[/tex]
61. [tex]\(5 x + 6 y = 4\)[/tex]
1. Express one variable in terms of the other from one of the equations. For example, solve equation [60] for [tex]\( y \)[/tex]:
[tex]\[y = \frac{16 - 12x}{3}\][/tex]
2. Substitute this expression into equation [61] and solve for [tex]\( x \)[/tex]:
[tex]\[5 x + 6 \left(\frac{16 - 12x}{3}\right) = 4\][/tex]
3. Simplify the equation:
[tex]\[5 x + 2 (16 - 12 x) = 4\][/tex]
[tex]\[5 x + 32 - 24 x = 4\][/tex]
[tex]\[-19 x + 32 = 4\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[-19 x = 4 - 32\][/tex]
[tex]\[-19 x = -28\][/tex]
[tex]\[x = \frac{28}{19}\][/tex]
5. Substitute [tex]\( x = \frac{28}{19} \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[y = \frac{16 - 12 \left(\frac{28}{19}\right)}{3}\][/tex]
[tex]\[y = \frac{16 - \frac{336}{19}}{3}\][/tex]
[tex]\[y = \frac{\frac{304}{19}}{3}\][/tex]
[tex]\[y = \frac{304}{57}\][/tex]
However, simplifying further:
[tex]\[y = -\frac{32}{57}\][/tex]
Thus, the solution to the first system is:
[tex]\[ x = \frac{28}{19}, \; y = -\frac{32}{57} \][/tex]
### System 2
62. [tex]\(4 x - y = 6\)[/tex]
[tex]\(-36 x + 12 = 9 y\)[/tex]
1. Solve equation 62.1 for [tex]\( y \)[/tex]:
[tex]\[ y = 4 x - 6 \][/tex]
2. Substitute this expression into equation 62.2:
[tex]\[ -36 x + 12 = 9 (4 x - 6) \][/tex]
3. Simplify the equation:
[tex]\[ -36 x + 12 = 36 x - 54 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 12 + 54 = 36 x + 36 x \][/tex]
[tex]\[ 66 = 72 x \][/tex]
[tex]\[ x = \frac{11}{12} \][/tex]
5. Substitute [tex]\( x = \frac{11}{12} \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = 4 \left(\frac{11}{12}\right) - 6 \][/tex]
[tex]\[ y = \frac{44}{12} - 6 \][/tex]
[tex]\[ y = \frac{44 - 72}{12} \][/tex]
[tex]\[ y = - \frac{28}{12} \][/tex]
[tex]\[ y = - \frac{7}{3} \][/tex]
Thus, the solution to the second system is:
[tex]\[ x = \frac{11}{12}, \; y = -\frac{7}{3} \][/tex]
### System 3
62. [tex]\(8 + 7 x = 10 y\)[/tex]
[tex]\( y = 3 x - 12\)[/tex]
1. Solve equation 62.3 for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{8 + 7 x}{10} \][/tex]
2. Substitute this expression into equation 62.4:
[tex]\[ \frac{8 + 7 x}{10} = 3 x - 12 \][/tex]
3. Multiply both sides by 10 to clear the fraction:
[tex]\[ 8 + 7 x = 30 x - 120 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 8 + 120 = 30 x - 7 x \][/tex]
[tex]\[ 128 = 23 x \][/tex]
[tex]\[ x = \frac{128}{23} \][/tex]
5. Substitute [tex]\( x = \frac{128}{23} \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = 3 \left(\frac{128}{23}\right) - 12 \][/tex]
[tex]\[ y = \frac{384}{23} - \frac{276}{23} \][/tex]
[tex]\[ y = \frac{108}{23} \][/tex]
Thus, the solution to the third system is:
[tex]\[ x = \frac{128}{23}, \; y = \frac{108}{23} \][/tex]
In summary, the solutions are:
1. [tex]\( x = \frac{28}{19}, \; y = -\frac{32}{57} \)[/tex]
2. [tex]\( x = \frac{11}{12}, \; y = -\frac{7}{3} \)[/tex]
3. [tex]\( x = \frac{128}{23}, \; y = \frac{108}{23} \)[/tex]
