To solve the system of equations using the elimination method, we aim to eliminate one of the variables by manipulating the equations. The given system is:

$-10x - 8y = 16 \quad \text{(1)}$ $6x + 3y = -15 \quad \text{(2)}.$

Step 1: Adjust coefficients to eliminate one variable

We decide to eliminate $y$. To do so, we need to make the coefficients of $y$ equal (with opposite signs). The least common multiple of $8$ and $3$ is $24$.

• Multiply (1) by $3$ to make the coefficient of $y$ in the first equation $-24$: $3(-10x - 8y) = 3(16) \quad \Rightarrow \quad -30x - 24y = 48 \quad \text{(3)}.$

• Multiply (2) by $8$ to make the coefficient of $y$ in the second equation $24$: $8(6x + 3y) = 8(-15) \quad \Rightarrow \quad 48x + 24y = -120 \quad \text{(4)}.$

Step 2: Add the equations to eliminate $y$

Add (3) and (4): $(-30x - 24y) + (48x + 24y) = 48 + (-120).$ $18x = -72.$

Step 3: Solve for $x$

$x = \frac{-72}{18} = -4.$

Step 4: Substitute $x = -4$ into one of the original equations

Substitute $x = -4$ into (2): $6(-4) + 3y = -15.$ $-24 + 3y = -15.$ $3y = 9.$ $y = \frac{9}{3} = 3.$

Final Solution

The solution to the system is: $x = -4, \quad y = 3.$

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Related Questions

1. How does the elimination method differ from substitution for solving systems of equations?
2. Can this system of equations be solved graphically? If so, how would the solution look on a graph?
3. What are some real-life applications of solving systems of equations?
4. How do you determine the most efficient method (elimination, substitution, or matrix) for solving a system?
5. How would the solution change if one of the equations were multiplied incorrectly during elimination?

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Tip:

Always double-check your coefficients when multiplying equations to avoid errors during elimination.