The equation you've provided is a linear equation in two variables, \(x\) and \(y\). Let's analyze and provide some possible solutions and interpretations.

### 1. Solving for One Variable in Terms of the Other

You can solve for one variable in terms of the other. Let's solve for \(y\) in terms of \(x\):

\[
10x + 5y = 12
\]

First, isolate \(y\) on one side:

\[
5y = 12 - 10x
\]

Now, divide by 5:

\[
y = \frac{12 - 10x}{5}
\]

### 2. Solving for \(x\) in Terms of \(y\)

Alternatively, you could solve for \(x\) in terms of \(y\):

\[
10x = 12 - 5y
\]

Now, divide by 10:

\[
x = \frac{12 - 5y}{10}
\]

### 3. Graphical Representation

This equation represents a straight line when graphed on a coordinate plane. You can rewrite the equation in slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept:

\[
y = -2x + \frac{12}{5}
\]

Here, the slope \(m = -2\) and the y-intercept is \(\frac{12}{5}\).

### 4. Finding Intercepts

- **x-intercept** (where \(y = 0\)):
  \[
  10x + 5(0) = 12 \implies x = \frac{12}{10} = 1.2
  \]

- **y-intercept** (where \(x = 0\)):
  \[
  10(0) + 5y = 12 \implies y = \frac{12}{5} = 2.4
  \]

### 5. General Solution Set

Since this is a linear equation with two variables, it has infinitely many solutions that lie on the line represented by this equation.

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Would you like more details or have any specific questions? Here are some related questions:

1. How do you find the point of intersection between two lines?
2. What is the method to find the slope of a line between two points?
3. How do you solve a system of linear equations with two variables?
4. How do you convert an equation to its slope-intercept form?
5. What are the practical applications of linear equations?

**Tip:** Graphing linear equations can help visualize the relationship between variables and find solutions quickly!

Explore how to solve the linear equation 10x + 5y = 12 with detailed explanations, including solving for one variable in terms of another, finding intercepts, and understanding the slope-intercept form. Suitable for students in Grades 6-8.

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