How to solve a system of two linear equations without graph. For example y = -4x+3 and y = 3x-11. Explain this 

To solve a system of two linear equations without using a graph, you can use the substitution method or the elimination method. Let's use the substitution method for the given system of equations:

1. \( y = -4x + 3 \)
2. \( y = 3x - 11 \)

### Substitution Method

Since both equations are equal to \( y \), you can set them equal to each other:

\[ -4x + 3 = 3x - 11 \]

Next, solve for \( x \):

1. Add \( 4x \) to both sides to get all the \( x \)-terms on one side:

\[ 3 = 7x - 11 \]

2. Add 11 to both sides to isolate the \( x \)-term:

\[ 14 = 7x \]

3. Divide both sides by 7 to solve for \( x \):

\[ x = 2 \]

Now that we have \( x \), substitute it back into one of the original equations to find \( y \). Let's use the first equation:

\[ y = -4(2) + 3 \]

\[ y = -8 + 3 \]

\[ y = -5 \]

So, the solution to the system of equations is \( (x, y) = (2, -5) \).

### Verification

To verify, substitute \( x = 2 \) into the second equation and check if \( y = -5 \):

\[ y = 3(2) - 11 \]

\[ y = 6 - 11 \]

\[ y = -5 \]

Since both equations give the same \( y \)-value for \( x = 2 \), the solution \( (2, -5) \) is correct.

Would you like more details or have any questions?

Here are 5 more questions you might ask:
1. How do you solve a system of linear equations using the elimination method?
2. What are some applications of systems of linear equations?
3. How can you solve a system of three linear equations?
4. What is the difference between consistent and inconsistent systems of equations?
5. How do you interpret the solution of a system of linear equations geometrically?

**Tip:** When solving systems of equations, always verify your solution by substituting back into the original equations to ensure accuracy.

Learn how to solve a system of two linear equations without using a graph. Step-by-step guide using the substitution method with detailed calculations for equations y = -4x + 3 and y = 3x - 11. Verify the solution accuracy. Suitable for Grades 7-9 students learning algebra.

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