  Math Tests - mathematics practice questions ## 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 variable equations with multipl./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\$\$

Express variable from equation (easy) ## Isolating the variable

Isolating a variable (a letter used as a placeholder for an unknown value, mostly x or y) means rearranging an equation so that a variable is on one side of the equation and everything else is on the other side.

The side of the equation that is on the left from the equal sign we call the left side (LS) and the other side is the 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 together to one side and isolate the variable on the other side.

Express y:
\$\$y+z =4\$\$

We will deduct z from both sides of the equation. Due to this z will be removed from the left side (LS) and deducted from 4 on the right side (RS).
\$\$y+z−z=4−z\$\$
\$\$y =4−z\$\$

Express y:
\$\$5y+x=11\$\$
\$\$5y+x−x=11−x\$\$
\$\$5y=11−x\$\$
\$\${5y}/5={11−x}/5\$\$
\$\$y={11−x}/5\$\$

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

Express x:
\$\$4x+12=q\$\$

To isolate x we have to divide both sides of equation by 4.
\$\${4x+12}/4=q/4\$\$
\$\$x+3=q/4\$\$
\$\$x+3−3=q/4−3\$\$
\$\$x=q/4−3\$\$

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

Express x:
\$\$4(x+3)=y\$\$

In this equation we can't just take 3 and deduct it from both sides. At first, we have to remove brackets.
\$\$4x+12=y\$\$
\$\$4x=y−12\$\$
\$\$x={y−12}/4\$\$
\$\$x=y/4−3\$\$

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

Express x:
\$\$q/{x+1}=4\$\$
\$\$q÷(x+1)=4\$\$
\$\$q=4(x+1)\$\$
\$\$q=4x+4\$\$
\$\$q−4=4x\$\$
\$\${q−4}/4=x\$\$
\$\$q/4−1=x\$\$

Multi-step equations with all the 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 the 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\$\$