5x + 4y=14  -4x-3y=-10 solve by equilivance

To solve the system of equations using the method of equivalence, we'll first rewrite each equation in a standard form and then express one variable in terms of the other.

The equations are:

1. \( 5x + 4y = 14 \)  
2. \( -4x - 3y = -10 \)  

**Step 1: Rearrange the first equation for \( y \):**

\[
4y = 14 - 5x
\]
\[
y = \frac{14 - 5x}{4}
\]

**Step 2: Substitute \( y \) in the second equation:**

\[
-4x - 3\left(\frac{14 - 5x}{4}\right) = -10
\]

Multiply through by 4 to eliminate the fraction:

\[
-16x - 3(14 - 5x) = -40
\]
\[
-16x - 42 + 15x = -40
\]
\[
-x - 42 = -40
\]
\[
-x = 2 \implies x = -2
\]

**Step 3: Substitute \( x \) back to find \( y \):**

\[
y = \frac{14 - 5(-2)}{4} = \frac{14 + 10}{4} = \frac{24}{4} = 6
\]

**Solution:**

The solution to the system of equations is:
\[
x = -2, \quad y = 6
\]

Would you like more details on any specific steps? Here are some questions to consider:

1. How can we check if this solution satisfies both equations?
2. What does the method of equivalence involve in general?
3. Could we solve this system using another method, like substitution or elimination?
4. How would the solution change if one equation were altered?
5. What do the graphical representations of these equations look like? 

**Tip:** Always verify your solution by substituting the values back into the original equations.

Discover how to solve the system of equations 5x + 4y = 14 and -4x - 3y = -10 using the method of equivalence. This step-by-step guide includes rearranging equations, substituting variables, and finding the solution x = -2 and y = 6. Ideal for students in Grades 9-11.

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