Solving and Verifying One-Variable Linear Equations

Hello again. Last lesson separated two ideas that are easy to confuse: simplifying an expression means cleaning it up, while solving an equation means finding the value of a variable that makes an equal sign true. You practiced the two skills that come first: distributing and combining like terms.

Today you will solve one-variable linear equations — משוואות לינאריות במשתנה אחד — and verify — לבדוק / לאמת — each answer by substitution — הצבה. A linear equation has a variable only to the first power, such as $3x+5=17$. By the end, you should be able to handle equations with numbers on both sides and equations with parentheses, while showing clear written steps for a test.

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An equation is a balance

The equal sign means that the left side and the right side have the same value. Think of it as a balanced scale: if you change only one side, it is no longer balanced.

To solve — לפתור — means to make the variable stand alone, or isolate the variable — לבודד את המשתנה.

The rule behind every valid algebra step is:

Do exactly the same operation to both sides of the equation.

For example:

$$x+7=15$$

The $7$ is added to $x$. Its inverse operation — פעולה הפוכה — is subtracting $7$. Subtract $7$ from both sides:

$$x+7-7=15-7$$

Simplify:

$$x=8$$

We did not simply “move” the $7$. We subtracted $7$ from both sides, which keeps the equation true.

Here are the inverse-operation pairs you will use most often:

What is attached to the variable?	Undo it by doing this to both sides
$+7$	subtract $7$
$-5$	add $5$
multiplied by $3$, as in $3x$	divide by $3$
multiplied by $-5$, as in $-5x$	divide by $-5$
divided by $4$, as in $\frac{x}{4}$	multiply by $4$

Useful vocabulary:

• both sides — שני אגפי המשוואה
• variable — משתנה
• coefficient — מקדם; the number multiplying a variable, such as $3$ in $3x$
• constant — קבוע; a number without a variable
• solution — פתרון; a value that makes the equation true
• isolate — לבודד
• inverse operation — פעולה הפוכה

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From one step to two steps

Some equations need only one inverse operation.

Example: a negative coefficient

Solve:

$$-5x=45$$

The variable is multiplied by $-5$, so divide both sides by $-5$:

$$\frac{-5x}{-5}=\frac{45}{-5}$$ $$x=-9$$

Be particularly careful here: a positive divided by a negative is negative.

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A two-step equation has both a constant and a coefficient attached to the variable.

Solve:

$$3x+5=17$$

First undo the addition of $5$. Subtract $5$ from both sides:

$$3x+5-5=17-5$$ $$3x=12$$

Now $x$ is multiplied by $3$. Divide both sides by $3$:

$$\frac{3x}{3}=\frac{12}{3}$$ $$x=4$$

The order matters. You cannot divide by $3$ at the beginning and leave the $5$ unchanged, because the $5$ is part of the entire left side. First remove the added or subtracted constant; then remove the coefficient.

A subtraction that looks different

Compare these two equations:

$$x-5=16$$ $$12-x=5$$

In the first, add $5$ to both sides:

$$x-5+5=16+5$$ $$x=21$$

In the second, the expression is $12-x$, which means $12+(-x)$. A safe method is to add $x$ to both sides:

$$12-x+x=5+x$$ $$12=5+x$$

Then subtract $5$ from both sides:

$$12-5=5+x-5$$ $$7=x$$

Write the answer in the usual order:

$$x=7$$

Subtraction is not reversible by merely changing the order: $12-x$ is not the same as $x-12$.

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The general method for linear equations

As equations become longer, do not guess the next move. Use the same dependable structure each time.

1. Simplify each side — פשט כל אגף. Distribute and combine like terms.
2. Put all variable terms on one side.
3. Put all constants on the other side.
4. Divide or multiply so that the coefficient of the variable becomes $1$.
5. Check by substitution in the original equation.

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Variables on both sides

Solve:

$$5x+4=8x-8$$

There are variable terms on both sides. Let the left side become the variable side. Subtract $5x$ from both sides:

$$5x+4-5x=8x-8-5x$$ $$4=3x-8$$

Now remove the $-8$ by adding $8$ to both sides:

$$4+8=3x-8+8$$ $$12=3x$$

Finally, divide both sides by $3$:

$$\frac{12}{3}=\frac{3x}{3}$$ $$4=x$$

Therefore:

$$x=4$$

You could instead decide to put variables on the right side, as we did here. Either side is acceptable. Choose the route that keeps signs manageable, and show each operation on both sides.

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Parentheses: simplify before solving

The distributive property and combining like terms from the previous lesson are now part of solving equations. Before isolating the variable, simplify each side.

Solve:

$$11(2x+3)=-3x+9(1+3x)$$

First distribute on both sides:

$$22x+33=-3x+9+27x$$

Combine the like terms on the right:

$$22x+33=24x+9$$

Now collect variable terms on the left by subtracting $24x$ from both sides:

$$22x-24x+33=24x-24x+9$$ $$-2x+33=9$$

Subtract $33$ from both sides:

$$-2x+33-33=9-33$$ $$-2x=-24$$

Divide both sides by $-2$:

$$\frac{-2x}{-2}=\frac{-24}{-2}$$ $$x=12$$

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The key order is:

1. Distribute every factor outside parentheses.
2. Combine like terms on each individual side.
3. Move variable terms and constants using equal operations on both sides.
4. Divide by the final coefficient.

A common error is to start “moving terms” while parentheses remain. Do not do that. Parentheses hide terms; reveal them first by distributing.

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Verify by substitution

A solution is not complete until it is checked. To substitute — להציב — means to replace the variable with your proposed answer.

Use the original equation, not only the last simplified line. For the previous example, we found:

$$x=12$$

Original equation:

$$11(2x+3)=-3x+9(1+3x)$$

Substitute $12$ for every $x$:

$$11(2(12)+3)=-3(12)+9(1+3(12))$$

Simplify the left side:

$$11(24+3)=11(27)=297$$

Simplify the right side:

$$-36+9(1+36)=-36+9(37)=-36+333=297$$

The result is:

$$297=297$$

This is a true statement, so $x=12$ is correct.

A substitution check catches common mistakes:

• a lost negative sign;
• an arithmetic mistake;
• distributing to only one term in parentheses;
• dividing by the wrong number;
• copying a term incorrectly.

Write a check in this compact test format:

$$\text{Check: }5(4)+4=8(4)-8$$ $$24=24$$

Therefore, $x=4$ is correct.

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When the variable disappears

Most linear equations in a basic test have one solution. But after simplifying, you may occasionally see the variable cancel completely.

Consider:

$$2(x+3)=2x+6$$

Distribute:

$$2x+6=2x+6$$

Subtract $2x$ from both sides:

$$6=6$$

This is always true, no matter which number replaces $x$. The equation has infinitely many solutions — אינסוף פתרונות.

Now consider:

$$3x+2=3x+5$$

Subtract $3x$ from both sides:

$$2=5$$

This is false. No value of $x$ can make it true, so there is no solution — אין פתרון.

These outcomes are not mistakes if your algebra steps were correct:

Final result	Meaning
$x=4$	One solution
$6=6$	Infinitely many solutions
$2=5$	No solution

Do not try to divide by the variable after it has disappeared. Read the final true or false statement instead.

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A reliable written-test layout

Keep one equation per line and show the operation clearly. For example:

$$4(2x-1)+3=5x+12$$

Distribute and simplify:

$$8x-4+3=5x+12$$ $$8x-1=5x+12$$

Subtract $5x$ from both sides:

$$3x-1=12$$

Add $1$ to both sides:

$$3x=13$$

Divide by $3$:

$$x=\frac{13}{3}$$

Even if a final answer is a fraction, it is still a perfectly valid solution. Do not change it to a decimal unless the question asks for a decimal approximation.

Before finishing any equation, use this checklist:

• Did I simplify both sides first?
• Did I distribute to every term inside parentheses?
• Did I do the same operation to both sides?
• Did I keep negative signs?
• Did I isolate the variable completely?
• Did I substitute my answer into the original equation and get a true statement?

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Key takeaways

• Solving an equation means finding the value that makes both sides equal.
• Treat an equation as a balance: every operation must be performed on both sides.
• Use inverse operations to isolate the variable.
• For longer equations, first simplify, then collect variables on one side and constants on the other.
• Verify — בדוק — by substituting the answer into the original equation.
• A final true number statement such as $6=6$ means infinitely many solutions; a false one such as $2=5$ means no solution.

Next, you will use your distributing and equation-reading skills in a different direction: factoring out the greatest common factor from algebraic expressions.