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]
