Get the information you need from a community of experts on IDNLearn.com. Ask anything and receive prompt, well-informed answers from our community of experienced experts.
Sagot :
Sure! Let's work through the simultaneous equations step-by-step.
### Equations for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
Given equations:
1. [tex]\(3a + b = 13\)[/tex]
2. [tex]\(6a - 7b = -10\)[/tex]
We need to solve these equations simultaneously. Here's how:
Step 1: Solve the first equation for [tex]\(b\)[/tex]:
[tex]\[ b = 13 - 3a \][/tex]
Step 2: Substitute this expression for [tex]\(b\)[/tex] into the second equation:
[tex]\[ 6a - 7(13 - 3a) = -10 \][/tex]
Step 3: Simplify and solve for [tex]\(a\)[/tex]:
[tex]\[ 6a - 91 + 21a = -10 \][/tex]
[tex]\[ 27a - 91 = -10 \][/tex]
[tex]\[ 27a = 81 \][/tex]
[tex]\[ a = 3 \][/tex]
Step 4: Substitute [tex]\(a = 3\)[/tex] back into the expression for [tex]\(b\)[/tex]:
[tex]\[ b = 13 - 3 \cdot 3 \][/tex]
[tex]\[ b = 13 - 9 \][/tex]
[tex]\[ b = 4 \][/tex]
So, the solutions for [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[ a = 3, \quad b = 4 \][/tex]
### Equations for [tex]\(p\)[/tex] and [tex]\(q\)[/tex]:
Given equations:
1. [tex]\(6p - 7q = 23\)[/tex]
2. [tex]\(18p - q = 89\)[/tex]
We need to solve these equations simultaneously. Here's how:
Step 1: Solve the second equation for [tex]\(q\)[/tex]:
[tex]\[ q = 18p - 89 \][/tex]
Step 2: Substitute this expression for [tex]\(q\)[/tex] into the first equation:
[tex]\[ 6p - 7(18p - 89) = 23 \][/tex]
Step 3: Simplify and solve for [tex]\(p\)[/tex]:
[tex]\[ 6p - 126p + 623 = 23 \][/tex]
[tex]\[ -120p + 623 = 23 \][/tex]
[tex]\[ -120p = -600 \][/tex]
[tex]\[ p = 5 \][/tex]
Step 4: Substitute [tex]\(p = 5\)[/tex] back into the expression for [tex]\(q\)[/tex]:
[tex]\[ q = 18 \cdot 5 - 89 \][/tex]
[tex]\[ q = 90 - 89 \][/tex]
[tex]\[ q = 1 \][/tex]
So, the solutions for [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are:
[tex]\[ p = 5, \quad q = 1 \][/tex]
### Equations for [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
Given equations:
1. [tex]\(5m - 2n = 20\)[/tex]
2. [tex]\(10m + 7n = 95\)[/tex]
We need to solve these equations simultaneously. Here's how:
Step 1: Solve the first equation for [tex]\(n\)[/tex]:
[tex]\[ 2n = 5m - 20 \][/tex]
[tex]\[ n = \frac{5m - 20}{2} \][/tex]
Step 2: Substitute this expression for [tex]\(n\)[/tex] into the second equation:
[tex]\[ 10m + 7\left(\frac{5m - 20}{2}\right) = 95 \][/tex]
Step 3: Simplify and solve for [tex]\(m\)[/tex]:
[tex]\[ 10m + \frac{35m - 140}{2} = 95 \][/tex]
[tex]\[ 10m + 17.5m - 70 = 95 \][/tex]
[tex]\[ 27.5m - 70 = 95 \][/tex]
[tex]\[ 27.5m = 165 \][/tex]
[tex]\[ m = 6 \][/tex]
Step 4: Substitute [tex]\(m = 6\)[/tex] back into the expression for [tex]\(n\)[/tex]:
[tex]\[ n = \frac{5 \cdot 6 - 20}{2} \][/tex]
[tex]\[ n = \frac{30 - 20}{2} \][/tex]
[tex]\[ n = \frac{10}{2} \][/tex]
[tex]\[ n = 5 \][/tex]
So, the solutions for [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are:
[tex]\[ m = 6, \quad n = 5 \][/tex]
### Summary:
[tex]\[ \begin{array}{ll} 3 a + b = 13 & a = 3 \\ 6 a - 7 b = -10 & b = 4 \\ 6 p - 7 q = 23 & p = 5 \\ 18 p - q = 89 & q = 1 \\ 5 m - 2 n = 20 & m = 6 \\ 10 m + 7 n = 95 & n = 5 \\ \end{array} \][/tex]
### Equations for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
Given equations:
1. [tex]\(3a + b = 13\)[/tex]
2. [tex]\(6a - 7b = -10\)[/tex]
We need to solve these equations simultaneously. Here's how:
Step 1: Solve the first equation for [tex]\(b\)[/tex]:
[tex]\[ b = 13 - 3a \][/tex]
Step 2: Substitute this expression for [tex]\(b\)[/tex] into the second equation:
[tex]\[ 6a - 7(13 - 3a) = -10 \][/tex]
Step 3: Simplify and solve for [tex]\(a\)[/tex]:
[tex]\[ 6a - 91 + 21a = -10 \][/tex]
[tex]\[ 27a - 91 = -10 \][/tex]
[tex]\[ 27a = 81 \][/tex]
[tex]\[ a = 3 \][/tex]
Step 4: Substitute [tex]\(a = 3\)[/tex] back into the expression for [tex]\(b\)[/tex]:
[tex]\[ b = 13 - 3 \cdot 3 \][/tex]
[tex]\[ b = 13 - 9 \][/tex]
[tex]\[ b = 4 \][/tex]
So, the solutions for [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[ a = 3, \quad b = 4 \][/tex]
### Equations for [tex]\(p\)[/tex] and [tex]\(q\)[/tex]:
Given equations:
1. [tex]\(6p - 7q = 23\)[/tex]
2. [tex]\(18p - q = 89\)[/tex]
We need to solve these equations simultaneously. Here's how:
Step 1: Solve the second equation for [tex]\(q\)[/tex]:
[tex]\[ q = 18p - 89 \][/tex]
Step 2: Substitute this expression for [tex]\(q\)[/tex] into the first equation:
[tex]\[ 6p - 7(18p - 89) = 23 \][/tex]
Step 3: Simplify and solve for [tex]\(p\)[/tex]:
[tex]\[ 6p - 126p + 623 = 23 \][/tex]
[tex]\[ -120p + 623 = 23 \][/tex]
[tex]\[ -120p = -600 \][/tex]
[tex]\[ p = 5 \][/tex]
Step 4: Substitute [tex]\(p = 5\)[/tex] back into the expression for [tex]\(q\)[/tex]:
[tex]\[ q = 18 \cdot 5 - 89 \][/tex]
[tex]\[ q = 90 - 89 \][/tex]
[tex]\[ q = 1 \][/tex]
So, the solutions for [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are:
[tex]\[ p = 5, \quad q = 1 \][/tex]
### Equations for [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
Given equations:
1. [tex]\(5m - 2n = 20\)[/tex]
2. [tex]\(10m + 7n = 95\)[/tex]
We need to solve these equations simultaneously. Here's how:
Step 1: Solve the first equation for [tex]\(n\)[/tex]:
[tex]\[ 2n = 5m - 20 \][/tex]
[tex]\[ n = \frac{5m - 20}{2} \][/tex]
Step 2: Substitute this expression for [tex]\(n\)[/tex] into the second equation:
[tex]\[ 10m + 7\left(\frac{5m - 20}{2}\right) = 95 \][/tex]
Step 3: Simplify and solve for [tex]\(m\)[/tex]:
[tex]\[ 10m + \frac{35m - 140}{2} = 95 \][/tex]
[tex]\[ 10m + 17.5m - 70 = 95 \][/tex]
[tex]\[ 27.5m - 70 = 95 \][/tex]
[tex]\[ 27.5m = 165 \][/tex]
[tex]\[ m = 6 \][/tex]
Step 4: Substitute [tex]\(m = 6\)[/tex] back into the expression for [tex]\(n\)[/tex]:
[tex]\[ n = \frac{5 \cdot 6 - 20}{2} \][/tex]
[tex]\[ n = \frac{30 - 20}{2} \][/tex]
[tex]\[ n = \frac{10}{2} \][/tex]
[tex]\[ n = 5 \][/tex]
So, the solutions for [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are:
[tex]\[ m = 6, \quad n = 5 \][/tex]
### Summary:
[tex]\[ \begin{array}{ll} 3 a + b = 13 & a = 3 \\ 6 a - 7 b = -10 & b = 4 \\ 6 p - 7 q = 23 & p = 5 \\ 18 p - q = 89 & q = 1 \\ 5 m - 2 n = 20 & m = 6 \\ 10 m + 7 n = 95 & n = 5 \\ \end{array} \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.
