Writing Linear Equations in Standard Form Guided Notes | Teaching ... - Free Printable

To write linear equations in Standard Form, we follow these rules:

> Standard Form of a linear equation is:
> Ax + By = C
> where A, B, and C are integers (no fractions or decimals),
> and A should be positive (if possible).

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Let’s solve each equation one by one.

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1.  (1/3)x + (1/2)y = -4

We need to eliminate fractions. The denominators are 3 and 2 → LCM = 6.

Multiply every term by 6:

6 × (1/3)x = 2x
6 × (1/2)y = 3y
6 × (-4) = -24

So:
→ 2x + 3y = -24

A = 2 (positive ✔), all integers ✔ → Done.

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2.  2x – 8y + 7 = 3

Move constants to the right side.

Subtract 7 from both sides:

2x – 8y = 3 – 7
→ 2x – 8y = -4

We can simplify this by dividing all terms by 2 (optional but cleaner):

→ x – 4y = -2

But standard form usually keeps integer coefficients with A positive — here A=1 already positive.

However, sometimes teachers prefer no common factors. Let’s check original instruction — it doesn’t specify simplifying, so either is acceptable. But let’s keep it as simplified unless told otherwise.

Actually, looking at the problem set, they likely want simplest integer form with A > 0.

So:
→ x – 4y = -2

Wait — but if we leave it as 2x – 8y = -4, that’s also correct, just not simplified. Since the goal is “standard form”, and standard form allows any integers as long as A ≥ 0, both are technically okay. But typically we reduce if possible.

Let me double-check: In many curricula, they require A, B, C to have no common factor other than 1. So yes, divide by 2.

Final: x – 4y = -2

But wait — what if they want A to be positive? It is. Good.

Alternatively, some might write it as:

Multiply by -1 to make constant positive? No — standard form does NOT require C to be positive. Only A should be non-negative.

So: x – 4y = -2 is fine.

But let’s see next problems for consistency.

Actually, let’s hold off and do all first, then decide.

Wait — perhaps better to avoid reducing if not necessary? Let’s look at example 3.

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3.  y + 4 = 2(x + 11)

First, expand the right side:

y + 4 = 2x + 22

Now move all variable terms to left, constants to right.

Subtract 2x from both sides:

-2x + y + 4 = 22

Subtract 4 from both sides:

-2x + y = 18

Now, we want A (coefficient of x) to be positive. Multiply entire equation by -1:

→ 2x – y = -18

Check: A=2>0, integers ✔ → Done.

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Back to #2: 2x – 8y = -4 → divide by 2 → x – 4y = -2

That’s simpler and still standard form. I’ll go with that.

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4.  y = (4/7)x + 5

Eliminate fraction. Denominator is 7 → multiply every term by 7:

7y = 4x + 35

Now move all terms to left side (or get Ax + By = C format):

Subtract 4x from both sides:

-4x + 7y = 35

Make A positive → multiply by -1:

→ 4x – 7y = -35

Check: A=4>0, integers ✔ → Done.

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5.  y = -(1/2)x – 1

Eliminate fraction. Multiply every term by 2:

2y = -x – 2

Add x to both sides:

x + 2y = -2

A=1>0, integers ✔ → Done.

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## Final Answers:

1.  (1/3)x + (1/2)y = -4 → 2x + 3y = -24

2.  2x – 8y + 7 = 3 → x – 4y = -2

3.  y + 4 = 2(x + 11) → 2x – y = -18

4. y = (4/7)x + 5 → 4x – 7y = -35

5.  y = -(1/2)x – 1 → x + 2y = -2

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Final Answer:
1. 2x + 3y = -24
2. x - 4y = -2
3. 2x - y = -18
4. 4x - 7y = -35
5. x + 2y = -2