Variable equations – Math Tests – Practice Exercises & Quizzes

Balancing the scales (easy)

Balancing the Scale

If the scale is level, the total weight (or value) on the left equals the total weight on the right.

In the picture:

Left pan: 4 identical green cubes (each has the same unknown value).
Right pan: two blocks labeled 12 and 12, so the total is 24.

Let one green cube be x. Because the scale is balanced, we can write the equation:

$$4x = 12 + 12 = 24$$

Now solve for x by dividing both sides by 4:

$$x = 24 ÷ 4 = 6$$

Answer: Each green cube has the value 6.

Remove equal weights

When a scale is balanced, you can remove the same items from both pans without breaking the balance.

Take away three green cubes and the block 4 from both sides. Because we removed the same weights from each pan, the scale stays balanced.

Let the green cube be x.

After removing:

Left pan: 1 green cube remains
Right pan: only the block 9 remains

So the simplified balance is:

$$x = 9$$

Answer: Each green cube has the value 9.

One-step equations (addition and subtraction)

Solving the equations

The most straightforward method to solve an equation is to isolate the variable (a letter used as a placeholder for an unknown value, mostly x or y) to one side of the equation and everything else to the other side.

All equations have two sides: a left side (LS) and a right side (RS). We can to do the same thing (adding, subtracting, multiplying every term, dividing every term) to both sides of the equation. This can help to bring like terms (numbers) together to one side and isolate the variable on the other side.

$$y+3 =8$$

We will deduct 3 from both sides of the equation. Due to this 3 will be removed from the left side (LS) and deducted from 8 on the right side (RS):

$$y+3−3=8−3$$

$$y =5$$

Other example:

$$5x+3=x+11$$

$$5x+3−3=x+11−3$$

$$5x=x+8$$

$$5x−x=x−x+8$$

$$4x=8$$

$$4x÷4=8÷4$$

$$x=2$$

Keep in mind that you have to multiply and divide every term.

$$12x+4=4x$$

To isolate x we have to divide both sides of equation by 4.

$${12x}/4+4/4={4x}/4$$

$$3x+1=x$$

$$3x−3x+1=x−3x$$

$$1=−2x$$

$$1/{−2}=−{1/2}=x$$

It is necessary to respect parenthesis as well. We need to simplify them before we do any operations with terms inside them.

$$4(x+3)=20$$

In this equation we can't just take 3 and deduct it from both sides. At first, we have to remove brackets.

$$4x+12=20$$

$$4x=20−12$$

$$4x=8$$

$$x=2$$

$$10−(x+4)=2$$

$$10−x−4=2$$

$$6−x=2$$

$$6−2=x$$

$$x=4$$

An expression in the denominator must be handled as it is an isolated expression in parentheses:

$$12/{x+1}=4$$

$$12÷(x+1)=4$$

$$12=4(x+1)$$

$$12=4x+4$$

$$12−4=4x$$

$$8=4x$$

$$x=2$$

One-step equations (multiplication and division)

Solving the equations

The most straightforward method to solve an equation is to isolate the variable (a letter used as a placeholder for an unknown value, mostly x or y) to one side of the equation and everything else to the other side.

All equations have two sides: a left side (LS) and a right side (RS). We can to do the same thing (adding, subtracting, multiplying every term, dividing every term) to both sides of the equation. This can help to bring like terms (numbers) together to one side and isolate the variable on the other side.

$$y+3 =8$$

We will deduct 3 from both sides of the equation. Due to this 3 will be removed from the left side (LS) and deducted from 8 on the right side (RS):

$$y+3−3=8−3$$

$$y =5$$

Other example:

$$5x+3=x+11$$

$$5x+3−3=x+11−3$$

$$5x=x+8$$

$$5x−x=x−x+8$$

$$4x=8$$

$$4x÷4=8÷4$$

$$x=2$$

Keep in mind that you have to multiply and divide every term.

$$12x+4=4x$$

To isolate x we have to divide both sides of equation by 4.

$${12x}/4+4/4={4x}/4$$

$$3x+1=x$$

$$3x−3x+1=x−3x$$

$$1=−2x$$

$$1/{−2}=−{1/2}=x$$

It is necessary to respect parenthesis as well. We need to simplify them before we do any operations with terms inside them.

$$4(x+3)=20$$

In this equation we can't just take 3 and deduct it from both sides. At first, we have to remove brackets.

$$4x+12=20$$

$$4x=20−12$$

$$4x=8$$

$$x=2$$

$$10−(x+4)=2$$

$$10−x−4=2$$

$$6−x=2$$

$$6−2=x$$

$$x=4$$

An expression in the denominator must be handled as it is an isolated expression in parentheses:

$$12/{x+1}=4$$

$$12÷(x+1)=4$$

$$12=4(x+1)$$

$$12=4x+4$$

$$12−4=4x$$

$$8=4x$$

$$x=2$$

Number pyramids with equations

Multi-step equations with all operations (easy)

Solving the equations

The most straightforward method to solve an equation is to isolate the variable (a letter used as a placeholder for an unknown value, mostly x or y) to one side of the equation and everything else to the other side.

All equations have two sides: a left side (LS) and a right side (RS). We can to do the same thing (adding, subtracting, multiplying every term, dividing every term) to both sides of the equation. This can help to bring like terms (numbers) together to one side and isolate the variable on the other side.

$$y+3 =8$$

We will deduct 3 from both sides of the equation. Due to this 3 will be removed from the left side (LS) and deducted from 8 on the right side (RS):

$$y+3−3=8−3$$

$$y =5$$

Other example:

$$5x+3=x+11$$

$$5x+3−3=x+11−3$$

$$5x=x+8$$

$$5x−x=x−x+8$$

$$4x=8$$

$$4x÷4=8÷4$$

$$x=2$$

Keep in mind that you have to multiply and divide every term.

$$12x+4=4x$$

To isolate x we have to divide both sides of equation by 4.

$${12x}/4+4/4={4x}/4$$

$$3x+1=x$$

$$3x−3x+1=x−3x$$

$$1=−2x$$

$$1/{−2}=−{1/2}=x$$

It is necessary to respect parenthesis as well. We need to simplify them before we do any operations with terms inside them.

$$4(x+3)=20$$

In this equation we can't just take 3 and deduct it from both sides. At first, we have to remove brackets.

$$4x+12=20$$

$$4x=20−12$$

$$4x=8$$

$$x=2$$

$$10−(x+4)=2$$

$$10−x−4=2$$

$$6−x=2$$

$$6−2=x$$

$$x=4$$

An expression in the denominator must be handled as it is an isolated expression in parentheses:

$$12/{x+1}=4$$

$$12÷(x+1)=4$$

$$12=4(x+1)$$

$$12=4x+4$$

$$12−4=4x$$

$$8=4x$$

$$x=2$$

Multi-step equations with all operations (difficult)

Solving the equations

The most straightforward method to solve an equation is to isolate the variable (a letter used as a placeholder for an unknown value, mostly x or y) to one side of the equation and everything else to the other side.

All equations have two sides: a left side (LS) and a right side (RS). We can to do the same thing (adding, subtracting, multiplying every term, dividing every term) to both sides of the equation. This can help to bring like terms (numbers) together to one side and isolate the variable on the other side.

$$y+3 =8$$

We will deduct 3 from both sides of the equation. Due to this 3 will be removed from the left side (LS) and deducted from 8 on the right side (RS).

$$y+3−3=8−3$$

$$y =5$$

$$5x+3=x+11$$

$$5x+3−3=x+11−3$$

$$5x=x+8$$

$$5x−x=x−x+8$$

$$4x=8$$

$$4x÷4=8÷4$$

$$x=2$$

Keep in mind that you have to multiply and divide every term.

$$12x+4=4x$$

To isolate x we have to divide both sides of equation by 4.

$${12x}/4+4/4={4x}/4$$

$$3x+1=x$$

$$3x−3x+1=x−3x$$

$$1=−2x$$

$$1/{−2}=−{1/2}=x$$

It is necessary to respect parenthesis as well. We need to simplify them before we do any operations with terms inside them.

$$4(x+3)=20$$

In this equation we can't just take 3 and deduct it from both sides. At first, we have to remove brackets.

$$4x+12=20$$

$$4x=20−12$$

$$4x=8$$

$$x=2$$

$$10−(x+4)=2$$

$$10−x−4=2$$

$$6−x=2$$

$$6−2=x$$

$$x=4$$

An expression in the denominator must be handled as it is an isolated expression in parentheses:

$$12/{x+1}=4$$

$$12÷(x+1)=4$$

$$12=4(x+1)$$

$$12=4x+4$$

$$12−4=4x$$

$$8=4x$$

$$x=2$$

Systems of equations (super easy)

System of Equations

♘ + ♘ + ♜ = 11
♜ + ♜ + ♘ + ♘ = 12

At first glance, this may look complicated: two knights and one rook equal 11, while two rooks and two knights equal 12.

Notice that we can rewrite the second equation as:

(♘ + ♘ + ♜) + ♜ = 12

From the first equation we already know that:

♘ + ♘ + ♜ = 11

So we replace this sum in parentheses (two knights and a rook) with the number 11:

11 + ♜ = 12

From this we immediately see that ♜ = 1, because 11 + 1 = 12.

Substitute back into the first equation:
♘ + ♘ + 1 = 11 → ♘ + ♘ = 10 →
♘ = 5

Result: knight ♘ = 5, rook ♜ = 1.

If one equation contains only a single symbol, the situation is simpler:

♘ + ♘ + ♜ = 13
♜ + ♜ + ♜ = 9

From the second equation it’s clear that three rooks equal 9. That means one rook has the value 3. Substitute this into the first equation:

♘ + ♘ + 3 = 13
♘ + ♘ = 10
♘ = 5