Today, let’s look at writing equations and then solving them mentally. But before we start trying to write equations, we probably should ask the question what is an equation. An equation is a sentence that contains two expressions separated by an equal sign.
Here’s an example of that: seven equals eight minus one. Here’s another example: eighteen equals thirteen plus 𝑥. This equation contains a variable which is 𝑥. This equation, eighteen equals thirteen plus 𝑥, is neither true or false until the variable is replaced with a number.
Let’s say we’ve replaced 𝑥 with a three. The equation is now false because eighteen is not equal to sixteen.
But when we’ve replaced the 𝑥 with a five, it becomes a true statement. We say that the equation here is true when 𝑥 equals five. We also say that five is the solution of the equation.
A solution of an equation is the numerical value for the variable that makes the sentence true. In this case, five makes our sentence true. Solving an equation is the process of finding a solution. When we’d try to figure out what the variable is equal to, we call that solving the equation.
Now let’s take a look at how we solve an equation with mental math. So here’s an equation: 𝑝 minus five equals twenty. Remember that we wanna solve these problems mentally without writing things down.
So we think some number take away five is equal to twenty, and that number must be five more than twenty. Therefore, 𝑝 must equal twenty-five. 𝑝 equals twenty-five is the correct answer.
Here’s another example: if 𝑘 equals sixty is twenty-eight plus twenty-six equal 𝑘 true or false? And again we wanna use a mental math strategy to solve this question. So you think about twenty-eight and twenty-six. They have an eight and a six in their ones position and sixty has a zero in its one position. eight plus six equals fourteen, and that means the solution to 𝑘 is going to have to have a four in the ones position.
At this point, we don’t know what 𝑘 is. But we know that if 𝑘 equals sixty, then this sentence must be false. We can say for sure that twenty-eight plus twenty-six does not equal sixty, which makes this statement false. Now we are going to look at a question that asks us to write an equation.
The total cost of a skateboard and knee pads is two hundred and forty-seven dollars. If the skateboard cost two hundred and thirty-five dollars, write an equation to find the cost of the knee pads 𝑘, then solve the equation. The problem is asking us to do two things: write an equation and then solve the equation that we write.
We’re given three different pieces of information: the total cost is two hundred and forty-seven dollars; the skateboard cost two hundred and thirty-five dollars; and the missing cost the cost that we don’t know is 𝑘, and that’s the cost of the knee pads.
We know that the total cost equals the skateboard price plus the knee pads price. Let’s substitute these two pieces with the information that we already know. Total cost is two hundred and forty-seven. Skateboard cost two hundred and thirty-five.
What should we put here for the knee pads price? Because we don’t know the knee pads price, we need to substitute a variable here. In our case, that’s where the 𝑘 goes.
Our equation becomes two hundred and forty-seven equals two hundred and thirty-five plus 𝑘. Step one of the problem is complete. This is an equation we can use to solve step two. And this part can be done mentally. What added to two hundred and thirty-five would equal two hundred and forty-seven?
Well if you add ten you get from two hundred and thirty-five to two hundred and forty-five. Add two more to that, and now you’re at two hundred and forty-seven. Put both of those numbers you added together ten and two, and you realise that 𝑘 must be equal to twelve. 𝑘 is equal to twelve, and that means the knee pads cost twelve dollars. Our final answer here is 𝑘 equals twelve dollars.