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Sagot :
To solve these systems of equations, we'll use the substitution method step by step.
### System 1:
[tex]\[ \begin{cases} 5x + y = 35 \\ 4x - y = 10 \end{cases} \][/tex]
1. Solve the first equation for [tex]\( y \)[/tex]:
[tex]\[ y = 35 - 5x \][/tex]
2. Substitute [tex]\( y = 35 - 5x \)[/tex] into the second equation:
[tex]\[ 4x - (35 - 5x) = 10 \][/tex]
3. Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 4x - 35 + 5x = 10 \][/tex]
[tex]\[ 9x - 35 = 10 \][/tex]
[tex]\[ 9x = 45 \][/tex]
[tex]\[ x = 5 \][/tex]
4. Substitute [tex]\( x = 5 \)[/tex] back into [tex]\( y = 35 - 5x \)[/tex]:
[tex]\[ y = 35 - 5(5) \][/tex]
[tex]\[ y = 10 \][/tex]
Solution 1: [tex]\( (5, 10) \)[/tex]
### System 2:
[tex]\[ \begin{cases} x - 7y = 69 \\ 10x - y = 0 \end{cases} \][/tex]
1. Solve the second equation for [tex]\( y \)[/tex]:
[tex]\[ y = 10x \][/tex]
2. Substitute [tex]\( y = 10x \)[/tex] into the first equation:
[tex]\[ x - 7(10x) = 69 \][/tex]
3. Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 70x = 69 \][/tex]
[tex]\[ -69x = 69 \][/tex]
[tex]\[ x = -1 \][/tex]
4. Substitute [tex]\( x = -1 \)[/tex] back into [tex]\( y = 10x \)[/tex]:
[tex]\[ y = 10(-1) \][/tex]
[tex]\[ y = -10 \][/tex]
Solution 2: [tex]\( (-1, -10) \)[/tex]
### System 3:
[tex]\[ \begin{cases} x - 16y = -36 \\ -x + y = 6 \end{cases} \][/tex]
1. Solve the second equation for [tex]\( x \)[/tex]:
[tex]\[ x = y - 6 \][/tex]
2. Substitute [tex]\( x = y - 6 \)[/tex] into the first equation:
[tex]\[ (y - 6) - 16y = -36 \][/tex]
3. Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[ y - 6 - 16y = -36 \][/tex]
[tex]\[ -15y - 6 = -36 \][/tex]
[tex]\[ -15y = -30 \][/tex]
[tex]\[ y = 2 \][/tex]
4. Substitute [tex]\( y = 2 \)[/tex] back into [tex]\( x = y - 6 \)[/tex]:
[tex]\[ x = 2 - 6 \][/tex]
[tex]\[ x = -4 \][/tex]
Solution 3: [tex]\( (-4, 2) \)[/tex]
### System 4:
[tex]\[ \begin{cases} x + y = -1 \\ x + 9y = -57 \end{cases} \][/tex]
1. Solve the first equation for [tex]\( x \)[/tex]:
[tex]\[ x = -1 - y \][/tex]
2. Substitute [tex]\( x = -1 - y \)[/tex] into the second equation:
[tex]\[ (-1 - y) + 9y = -57 \][/tex]
3. Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[ -1 - y + 9y = -57 \][/tex]
[tex]\[ -1 + 8y = -57 \][/tex]
[tex]\[ 8y = -56 \][/tex]
[tex]\[ y = -7 \][/tex]
4. Substitute [tex]\( y = -7 \)[/tex] back into [tex]\( x = -1 - y \)[/tex]:
[tex]\[ x = -1 - (-7) \][/tex]
[tex]\[ x = -1 + 7 \][/tex]
[tex]\[ x = 6 \][/tex]
Solution 4: [tex]\( (6, -7) \)[/tex]
Thus, the Ordered Pairs for the systems of equations are:
1. [tex]\( (5, 10) \)[/tex]
2. [tex]\( (-1, -10) \)[/tex]
3. [tex]\( (-4, 2) \)[/tex]
4. [tex]\( (6, -7) \)[/tex]
### System 1:
[tex]\[ \begin{cases} 5x + y = 35 \\ 4x - y = 10 \end{cases} \][/tex]
1. Solve the first equation for [tex]\( y \)[/tex]:
[tex]\[ y = 35 - 5x \][/tex]
2. Substitute [tex]\( y = 35 - 5x \)[/tex] into the second equation:
[tex]\[ 4x - (35 - 5x) = 10 \][/tex]
3. Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 4x - 35 + 5x = 10 \][/tex]
[tex]\[ 9x - 35 = 10 \][/tex]
[tex]\[ 9x = 45 \][/tex]
[tex]\[ x = 5 \][/tex]
4. Substitute [tex]\( x = 5 \)[/tex] back into [tex]\( y = 35 - 5x \)[/tex]:
[tex]\[ y = 35 - 5(5) \][/tex]
[tex]\[ y = 10 \][/tex]
Solution 1: [tex]\( (5, 10) \)[/tex]
### System 2:
[tex]\[ \begin{cases} x - 7y = 69 \\ 10x - y = 0 \end{cases} \][/tex]
1. Solve the second equation for [tex]\( y \)[/tex]:
[tex]\[ y = 10x \][/tex]
2. Substitute [tex]\( y = 10x \)[/tex] into the first equation:
[tex]\[ x - 7(10x) = 69 \][/tex]
3. Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 70x = 69 \][/tex]
[tex]\[ -69x = 69 \][/tex]
[tex]\[ x = -1 \][/tex]
4. Substitute [tex]\( x = -1 \)[/tex] back into [tex]\( y = 10x \)[/tex]:
[tex]\[ y = 10(-1) \][/tex]
[tex]\[ y = -10 \][/tex]
Solution 2: [tex]\( (-1, -10) \)[/tex]
### System 3:
[tex]\[ \begin{cases} x - 16y = -36 \\ -x + y = 6 \end{cases} \][/tex]
1. Solve the second equation for [tex]\( x \)[/tex]:
[tex]\[ x = y - 6 \][/tex]
2. Substitute [tex]\( x = y - 6 \)[/tex] into the first equation:
[tex]\[ (y - 6) - 16y = -36 \][/tex]
3. Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[ y - 6 - 16y = -36 \][/tex]
[tex]\[ -15y - 6 = -36 \][/tex]
[tex]\[ -15y = -30 \][/tex]
[tex]\[ y = 2 \][/tex]
4. Substitute [tex]\( y = 2 \)[/tex] back into [tex]\( x = y - 6 \)[/tex]:
[tex]\[ x = 2 - 6 \][/tex]
[tex]\[ x = -4 \][/tex]
Solution 3: [tex]\( (-4, 2) \)[/tex]
### System 4:
[tex]\[ \begin{cases} x + y = -1 \\ x + 9y = -57 \end{cases} \][/tex]
1. Solve the first equation for [tex]\( x \)[/tex]:
[tex]\[ x = -1 - y \][/tex]
2. Substitute [tex]\( x = -1 - y \)[/tex] into the second equation:
[tex]\[ (-1 - y) + 9y = -57 \][/tex]
3. Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[ -1 - y + 9y = -57 \][/tex]
[tex]\[ -1 + 8y = -57 \][/tex]
[tex]\[ 8y = -56 \][/tex]
[tex]\[ y = -7 \][/tex]
4. Substitute [tex]\( y = -7 \)[/tex] back into [tex]\( x = -1 - y \)[/tex]:
[tex]\[ x = -1 - (-7) \][/tex]
[tex]\[ x = -1 + 7 \][/tex]
[tex]\[ x = 6 \][/tex]
Solution 4: [tex]\( (6, -7) \)[/tex]
Thus, the Ordered Pairs for the systems of equations are:
1. [tex]\( (5, 10) \)[/tex]
2. [tex]\( (-1, -10) \)[/tex]
3. [tex]\( (-4, 2) \)[/tex]
4. [tex]\( (6, -7) \)[/tex]
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