Solving a system of equations by elimination and substitution, resulting in the solution (1.25, 2.5).

You’ve uploaded an image showing a step-by-step solution to a system of linear equations using the elimination method, followed by substitution and verification. Let’s walk through the problem, explain each step clearly, and confirm the solution is correct.

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## 🧩 The System of Equations

We are given:

> Equation 1:      `3x + 3y = 11.25`
> Equation 2:      `4x + 2y = 10`

We want to find values of `x` and `y` that satisfy both equations simultaneously.

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## ✔ Step 1: Eliminate One Variable (Elimination Method)

The goal is to add or subtract the equations to eliminate one variable. To do this, we make the coefficients of one variable opposites (e.g., +6y and -6y).

Multiply Equation 1 by 2:

> `2 × (3x + 3y) = 2 × 11.25`
> → `6x + 6y = 22.50`

Multiply Equation 2 by -3:

> `-3 × (4x + 2y) = -3 × 10`
> → `-12x - 6y = -30`

Now write them together:

```
6x + 6y = 22.50
-12x - 6y = -30
------------------
```

Add them vertically:

- `6x + (-12x) = -6x`
- `6y + (-6y) = 0`
- `22.50 + (-30) = -7.50`

So we get:

> -6x = -7.50

Divide both sides by -6:

> x = (-7.50) / (-6) = 1.25

✔ So, x = 1.25

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## ✔ Step 2: Substitute x Back to Find y

Now plug `x = 1.25` into one of the original equations. The image uses Equation 2: `4x + 2y = 10`

Substitute:

> `4(1.25) + 2y = 10`
> → `5 + 2y = 10`

Subtract 5 from both sides:

> `2y = 5`

Divide by 2:

> y = 2.5

✔ So, y = 2.5

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## ✔ Step 3: Check the Solution in Both Equations

We now verify that `(x, y) = (1.25, 2.5)` satisfies both original equations.

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🔍 Check in Equation 1: `3x + 3y = 11.25`

> `3(1.25) + 3(2.5) = ?`
> → `3.75 + 7.5 = 11.25` ✔️

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🔍 Check in Equation 2: `4x + 2y = 10`

> `4(1.25) + 2(2.5) = ?`
> → `5 + 5 = 10` ✔️

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## 🎯 Final Answer:

> The solution to the system is `(x, y) = (1.25, 2.5)`

This means the two lines represented by the equations intersect at the point `(1.25, 2.5)`.

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## 💡 Key Takeaways:

- Elimination Method: Multiply equations to create opposite coefficients, then add to eliminate a variable.
- Substitution: Once you have one variable, plug it back into any equation to solve for the other.
- Verification: Always check your solution in both original equations to ensure accuracy.

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✔ Your work in the image is perfectly correct — great job following the steps! The solution `(1.25, 2.5)` is accurate and verified.

Let me know if you’d like to see how to solve this using graphing or substitution from the start!