Solving equations (OCR GCSE Maths): Revision Notes

There are many different ways to solve equations, and if you're happy with how you have been taught, stick to it. We are going to show you a method we like…

• Example: If you add $5$ to a number, you subtract $5$ to reverse it.

• Example: If you multiply a number by $3$ , you divide by $3$ to reverse it.

Example 1: Solving a Linear Equation

Problem: Solve $7p−3=32.$

Solution:

• Step 1: Identify the operations applied to $p$.
• $p$ is first multiplied by $7$, then $3$ is subtracted.
• Step 2: Start unwrapping the variable using inverse operations. Begin with the last operation (subtraction) and reverse it by adding $3$ to both sides.

• Step 3: Now, divide both sides by $7$ (the inverse of multiplying by $7$):

• Step 4: Check your solution by substituting $p=5$ back into the original equation:

• Since both sides equal $32$, the solution is correct.
• Final Answer:

Example 2: Solving an Equation with Brackets

Problem: Solve $2(3r+6)=36$.

Step-by-Step Solution:

Step 1: Expand the Brackets

Before solving the equation, we need to expand the brackets to simplify the equation.

So the equation becomes:

Step 2: Isolate the Variable

To solve for $r$, we need to isolate the variable on one side of the equation. Start by reversing the operations applied to $r$.

• Subtract $12$ from both sides:

This simplifies to:

• Divide both sides by $6$:

This simplifies to:

Step 3: Check the Solution

It's always a good idea to check your solution by substituting the value of $r$ back into the original equation to ensure it balances.

Substitute $r=4$ into the original equation:

Substituting $r=4$:

Since both sides equal $36$, the solution $r=4$ is correct.

Example 3: Solving an Equation with a Fraction

Problem: Solve $6+\frac{k}{5}=−1$.

Step-by-Step Solution:

Step 1: Identify the Structure of the Equation

• The unknown letter $k$ only appears on the left-hand side of the equation.
• There is no negative sign in front of $k$, and $k$ is not in the denominator of the fraction. This makes the equation easier to handle.

Step 2: Understand the Order of Operations

• The equation $6+\frac{5}{k}=−1$ involves adding $6$ to $5k$.
• To solve for $k$, we must reverse the operations applied to it, starting with the last operation according to BODMAS.

Step 3: Apply Inverse Operations

• Subtract $6$ from both sides to begin isolating the variable:

Simplifies to:

• Multiply both sides by $5$ to eliminate the fraction:

Simplifies to:

Step 4: Check the Solution

It's crucial to check your answer by substituting $k=−35$ back into the original equation to ensure it balances.

Substitute $k=−35$ into the original equation:

Since both sides equal $−1$, the solution $k=−35$ is correct.

Example 4: Solving an Equation with a Negative Coefficient

Problem: Solve $24−3m=6$

Step-by-Step Solution:

Step 1: Identify the Structure of the Equation

• The unknown letter $m$ appears on the left-hand side of the equation.
• Notice that $m$ has a negative coefficient $−3m$, which can make solving the equation slightly more complex. However, by using inverse operations, we can handle it easily.

Step 2: Use Inverse Operations to Simplify

• To eliminate the negative sign in front of $m$, add $3m$ to both sides of the equation:

This simplifies to:

Now, the equation is easier to manage as the variable $m$ has a positive coefficient.

Step 3: Isolate the Variable

• Subtract $6$ from both sides to move the constant term to the left-hand side:

This simplifies to:

• Divide both sides by $3$ to solve for $m$:

This simplifies to:

Step 4: Check the Solution

It's crucial to check your answer by substituting $m=6$ back into the original equation to ensure it balances.

Substitute $m=6$ into the original equation:

Since both sides equal $6$, the solution $m=6$ is correct.

Example 5: Solving an Equation with Variables on Both Sides

Problem: Solve $7y+3=10y−6$.

Step-by-Step Solution:

Step 1: Collect the Variables on One Side

• Start by moving all the terms involving the variable y to one side of the equation. To do this, subtract $7y$from both sides:

This simplifies to:

Now, all the y terms are on the right side, and the equation is easier to solve.

Step 2: Use Inverse Operations to Isolate the Variable

• To isolate $y$, start by adding $6$ to both sides to move the constant term on the right-hand side:

This simplifies to:

• Next, divide both sides by $3$ to solve for $y$:

​

This simplifies to:

Step 3: Check the Solution

It's always important to check your answer by substituting $y=3$ back into the original equation to ensure it balances.

Substitute $y=3$ into the original equation:

Left-hand side:

Right-hand side:

Since both sides equal $24$, the solution $y=3$ is correct.

Example 6: Solving an Equation with the Variable in the Denominator

Problem: Solve .

Step-by-Step Solution:

Step 1: Eliminate the Fraction

• The variable $g$ is in the denominator, which makes the equation slightly more complex. To eliminate the fraction, multiply both sides of the equation by the denominator $g−1$:

This simplifies to:

Now the equation is simpler, and we no longer have a fraction.

Step 2: Expand the Brackets

• Expand the right-hand side of the equation by multiplying out the bracket:

Step 3: Use Inverse Operations to Isolate the Variable

• To solve for $g$, start by adding $5$ to both sides to move the constant term on the right-hand side:

This simplifies to:

• Next, divide both sides by $5$ to solve for $g:$

This simplifies to:

Step 4: Check the Solution

It's important to check your answer by substituting $g=6$ back into the original equation to ensure it balances.

Substitute $g=6$ into the original equation:

Since both sides equal $5$, the solution $g=6$ is correct.