https://file.aiquickdraw.com/mathbot/file/17274724017165la03t5c.jpg

The image shows a system of two equations:

\[
\begin{align*}
9x - 4y &= -23 \\
5x - 3y &= -19
\end{align*}
\]

We need to solve for \(x\) and \(y\). Let's solve this system step by step using the method of elimination. I'll begin by eliminating one of the variables.

### Step 1: Eliminate \(y\)
To eliminate \(y\), we want the coefficients of \(y\) in both equations to be equal. We can multiply the first equation by 3 and the second equation by 4, so that the coefficients of \(y\) will both be \(-12\).

1. Multiply the first equation by 3:
   \[
   3(9x - 4y) = 3(-23)
   \]
   which simplifies to:
   \[
   27x - 12y = -69
   \]

2. Multiply the second equation by 4:
   \[
   4(5x - 3y) = 4(-19)
   \]
   which simplifies to:
   \[
   20x - 12y = -76
   \]

### Step 2: Subtract the equations
Now, subtract the second equation from the first to eliminate \(y\):
\[
(27x - 12y) - (20x - 12y) = -69 - (-76)
\]
Simplifying:
\[
7x = 7
\]
Solving for \(x\):
\[
x = 1
\]

### Step 3: Substitute \(x = 1\) into one of the original equations
Substitute \(x = 1\) into the second equation:
\[
5(1) - 3y = -19
\]
Simplifying:
\[
5 - 3y = -19
\]
Subtract 5 from both sides:
\[
-3y = -24
\]
Solving for \(y\):
\[
y = 8
\]

Thus, the solution is:
\[
x = 1, \quad y = 8
\]

Let me know if you'd like more details or further clarification!

Here are five related questions to explore further:
1. Can you solve the same system using substitution instead of elimination?
2. What happens when two systems have no solution or infinite solutions?
3. How can you check if the solution \(x = 1\) and \(y = 8\) is correct?
4. What are the geometric interpretations of the solution to a system of equations?
5. How would the system change if one equation were a multiple of the other?

**Tip**: Always double-check your solution by substituting the values back into the original equations to verify correctness.

Solve the following system of equations:
9x - 4y = -23
5x - 3y = -19

Learn how to solve the system of linear equations 9x - 4y = -23 and 5x - 3y = -19 using the elimination method. This detailed solution includes steps to eliminate variables, solve for x and y, and verify the solution. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation