Solve this system of equations using substitution method.Must show work/steps on your work page {𝑦=3𝑥−44𝑥+3𝑦=1(, )

1. Break Down the Problem

We need to solve the system of equations given by:

1. $y = 3x - 4$
2. $4x + 3y = 1$

We will use the substitution method, which involves substituting the expression for $y$ from the first equation into the second equation.

2. Relevant Concepts

The substitution method entails the following steps:

• Isolate one variable in one of the equations.
• Substitute that expression into the other equation.
• Solve for the remaining variable.

3. Analysis and Detail

Step 1: Substitute $y$

We take the expression for $y$ from the first equation and substitute it into the second equation: $4x + 3(3x - 4) = 1$

Step 2: Simplify the Equation

Now, distribute the $3$: $4x + 9x - 12 = 1$

Combine like terms: $13x - 12 = 1$

Step 3: Solve for $x$

Now we isolate $x$: $13x = 1 + 12$ $13x = 13$ $x = 1$

Step 4: Find $y$

Now that we have $x$, we can use the first equation to find $y$: $y = 3(1) - 4$ $y = 3 - 4$ $y = -1$

4. Verify and Summarize

We found $x = 1$ and $y = -1$. Let's verify these values by substituting them back into the original equations:

1. For $y = 3x - 4$: $-1 = 3(1) - 4 \quad \text{(True)}$
2. For $4x + 3y = 1$: $4(1) + 3(-1) = 1 \quad \text{(True)}$

Both equations hold true.

Final Answer

The solution to the system of equations is: $x = 1, \quad y = -1$