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Ontario Tech acknowledges the lands and people of the Mississaugas of Scugog Island First Nation.

We are thankful to be welcome on these lands in friendship. The lands we are situated on are covered by the Williams Treaties and are the traditional territory of the Mississaugas, a branch of the greater Anishinaabeg Nation, including Algonquin, Ojibway, Odawa and Pottawatomi. These lands remain home to many Indigenous nations and peoples.

We acknowledge this land out of respect for the Indigenous nations who have cared for Turtle Island, also called North America, from before the arrival of settler peoples until this day. Most importantly, we acknowledge that the history of these lands has been tainted by poor treatment and a lack of friendship with the First Nations who call them home.

This history is something we are all affected by because we are all treaty people in Canada. We all have a shared history to reflect on, and each of us is affected by this history in different ways. Our past defines our present, but if we move forward as friends and allies, then it does not have to define our future.

Learn more about Indigenous Education and Cultural Services

Introduction to Solving Equations

Let’s review some background material that you’ll need to study equation solving.

Definition of an Equation

An equation is a statement of equality between two expressions separated into a left and right side by an equal sign (=). Inequalities are algebraic relations showing that a quantity is greater than (>), greater than or equal to (≥), less than (<), or less than or equal to (≤) another quantity.

What It Means to Solve an Equation

Finding a solution to an equation means finding the set of all x values that satisfy the equation when substituted into it. With this in mind, to check that a given value is indeed a solution, substitute it into the equation and make sure that the left-hand side (LHS) of the equation has the same value as the right-hand side (RHS).

Example: Solve for x in 3x – 5 = 1 and check your answer.

Solution:

      3x – 5 = 1

3x – 5 + 5 = 1 + 5

              3x = 6

          3x/3 = 6/3

                 x = 2

Now, let’s check that x = 2 is indeed a solution by verifying that LHS = RHS

LHS                            RHS

3(2) - 5                           1

= 6 - 5

= 1

Since LHS = RHS, we conclude that 2 is indeed a solution.

What it means to find a solution to an inequality

A solution to an inequality means finding the set of all x values that satisfy the required inequality when substituted in.

Example: Solve for the set of x that satisfiesx - 2  > 10.

   x - 2   > 10

(√ x - 2  )> 102

         x – 2 > 100

                x > 102

It’s a bit harder to check our answer here, but you’ll notice that if you substitute in a value of x that’s greater than 102 (e.g., 146), the inequality should hold (i.e., be satisfied), but if you substitute in a value of x smaller than 102 (e.g., 83), the inequality isn’t satisfied. This doesn’t prove that your answer is right, but it does give you a bit more confidence in it, and it might help you to find out if your answer is wrong.

Collecting like terms

The first step in solving an equation is usually to collect like terms. This may include adding and subtracting terms from each side to get all like terms on one side of an equation. Once the like terms have been collected it becomes possible to solve an equation by isolating the variable using multiplication, division, or other operations, and to finally solve for the unknown.

Example: Solve 3x + 5 = 7 – x for x.

Solution:

     3x + 5 = 7 – x

3x + 5 - 5 = 7 – x – 5

             3x = 2 – x

       3x + x = 2 – x + x

             4x = 2

         4x/4 = 2/4

                x = 1/2