https://file.aiquickdraw.com/mathbot/file/17291116891774eqd9akx.jpg

Let's solve the system of linear equations provided in the image:

\[
\begin{aligned}
4b + 7a &= 90 \tag{1} \\
5a - 6b &= 20 \tag{2}
\end{aligned}
\]

We'll use the **substitution or elimination method** to find the values of \(a\) and \(b\). Let's proceed with the elimination method.

---

### Step 1: Rearrange both equations
Let's rewrite the system for convenience:

1. \(4b + 7a = 90\)
2. \(-6b + 5a = 20\)

---

### Step 2: Eliminate one variable
We aim to eliminate one variable by multiplying the equations so that the coefficients of either \(b\) or \(a\) match. Let’s eliminate \(b\) by multiplying the first equation by 6 and the second equation by 4:

\[
6(4b + 7a) = 6(90) \quad \Rightarrow \quad 24b + 42a = 540
\]
\[
4(-6b + 5a) = 4(20) \quad \Rightarrow \quad -24b + 20a = 80
\]

Now, add the two equations to eliminate \(b\):

\[
(24b - 24b) + (42a + 20a) = 540 + 80
\]
\[
62a = 620
\]

---

### Step 3: Solve for \(a\)
\[
a = \frac{620}{62} = 10
\]

---

### Step 4: Substitute \(a = 10\) into one of the original equations
Let’s substitute \(a = 10\) into the first equation \(4b + 7a = 90\):

\[
4b + 7(10) = 90
\]
\[
4b + 70 = 90
\]
\[
4b = 90 - 70 = 20
\]
\[
b = \frac{20}{4} = 5
\]

---

### Step 5: Write the solution
The solution to the system is:
\[
a = 10, \quad b = 5
\]

---

Would you like more details or have further questions?

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Here are 5 related questions for further exploration:
1. How do you check if the solution to a system of equations is correct?
2. What are the advantages of the elimination method over substitution?
3. How do you solve systems with more than two variables?
4. What is the graphical interpretation of solving systems of linear equations?
5. How do you use matrices to solve systems of equations?

**Tip:** Always verify your solution by substituting it back into the original equations. This ensures no mistakes were made in the algebraic steps.

Solve the system of equations: 4b + 7a = 90, 5a - 6b = 20

Learn how to solve the system of linear equations 4b + 7a = 90 and 5a - 6b = 20 using the elimination method. This step-by-step solution covers algebraic concepts suitable for students in Grades 9-11.

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