Certainly! Let's go over solving systems of linear equations using elimination and substitution.

We will first clarify what each method involves, then we'll combine them. Typically, elimination is a method to get rid of one variable by adding or subtracting equations, while substitution involves solving one equation for one variable and then substituting that into the other equation.

Let’s break this down step-by-step with a simple example.

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Example System of Equations:

$3x + 2y = 16 \quad \text{(Equation 1)}$ $4x - 2y = 10 \quad \text{(Equation 2)}$

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Step 1: Choose One Equation to Solve for One Variable

Substitution comes into play here. Pick either equation and solve it for one variable in terms of the other.

Let’s solve Equation 1 for $x$: $3x + 2y = 16$ Solve for $x$: $3x = 16 - 2y$ $x = \frac{16 - 2y}{3}$

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Step 2: Substitute the Expression into the Other Equation

Now, we will substitute the expression for $x$ into Equation 2.

Start with Equation 2: $4x - 2y = 10$

Substitute $x = \frac{16 - 2y}{3}$ into this: $4\left(\frac{16 - 2y}{3}\right) - 2y = 10$

Simplify the equation: $\frac{4(16 - 2y)}{3} - 2y = 10$ $\frac{64 - 8y}{3} - 2y = 10$

Multiply through by 3 to eliminate the fraction: $64 - 8y - 6y = 30$ $64 - 14y = 30$

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Step 3: Solve for the Second Variable

Now, solve for $y$: $64 - 14y = 30$ Subtract 64 from both sides: $-14y = 30 - 64$ $-14y = -34$ Divide both sides by $-14$: $y = \frac{-34}{-14} = \frac{17}{7}$

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Step 4: Substitute $y$ Back Into the First Equation

Now that we have $y = \frac{17}{7}$, we substitute this back into the equation we found for $x$.

Recall: $x = \frac{16 - 2y}{3}$ Substitute $y = \frac{17}{7}$: $x = \frac{16 - 2\left(\frac{17}{7}\right)}{3}$ Simplify: $x = \frac{16 - \frac{34}{7}}{3}$ Convert 16 to have a denominator of 7: $x = \frac{\frac{112}{7} - \frac{34}{7}}{3}$ $x = \frac{\frac{78}{7}}{3}$ Now simplify: $x = \frac{78}{21} = \frac{26}{7}$

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Final Answer:

So, the solution to the system of equations is: $x = \frac{26}{7}, \quad y = \frac{17}{7}$

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Key Steps in Summary:

1. Solve one equation for one variable (usually $x$ or $y$).
2. Substitute this expression into the other equation.
3. Solve for the second variable.
4. Substitute the value of the second variable back into the first equation to find the first variable.

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Would you like more details on any of these steps or have further questions?

Here are 5 related questions to deepen understanding:

1. How do you solve a system using elimination instead of substitution?
2. What happens if the coefficients are fractions in elimination?
3. How can you recognize when a system has no solution or infinite solutions?
4. What if both equations in a system are already solved for $y$ or $x$?
5. What is the geometric interpretation of a system of equations with one solution?

Tip: If one equation is already close to being solved for one variable, substitution becomes very straightforward!