![]() Remember, we want to set this expression equal to zero. Let’s collect like terms on the left-hand side first. Then, our earlier equation becomes □ squared plus this expression plus the result of distributing negative seven □ across negative nine □ minus seven. And this simplifies to 81□ squared plus 126□ plus 49. Finally, negative seven times negative seven is 49. ![]() Next, negative nine □ times negative seven and negative seven times negative nine □ gives us two lots of 63□. Then, we multiply the first term in each expression to get 81□ squared. That’s negative nine □ minus seven multiplied by itself. Let’s multiply out the expression negative nine □ minus seven squared. And so we’re probably going to end up using the quadratic formula.īefore we do though, let’s distribute the parentheses and set this equation equal to zero. That is an indication to us that the quadratic equation will not be easily solved by factoring. It tells us to give values to two decimal places. There is a hint as to how we’re going to solve that quadratic equation in the question. And we might notice that when we distribute any parentheses, we’re going to have a quadratic equation in terms of □. And so when we do, equation two becomes □ squared plus negative nine □ minus seven squared minus seven □ times negative nine □ minus seven equals four. And to do so, we simply replace every instance of □ with the expression negative nine □ minus seven and then manipulate using the order of operations. Now, if we define this equation to be equation one and our quadratic to be equation two, we’re going to substitute equation one into equation two. And we see that this equation can be alternatively written as □ equals negative nine □ minus seven. To do so, we’re going to subtract nine □ and seven from both sides of the equation. It follows that since the coefficient of □ is one, in other words there is simply one □ in the equation, we should make □ the subject. ![]() We want to find a way to make either □ or □ the subject. So let’s inspect this first equation: □ plus nine □ plus seven equals zero. We’re going to form an expression for either □ or □ with our first equation and then substitute it into our second. And when we solve these systems of equations, the most sensible method to choose is substitution. This involves □ squared and □ squared terms. That’s an equation whose graph is simply a straight line. However, we have one linear equation: □ plus nine □ plus seven equals zero. There are a number of techniques we can use to solve a system of linear equations. ![]() Find all of the solutions to the simultaneous equations □ plus nine □ plus seven equals zero and □ squared plus □ squared minus seven □□ equals four. ![]()
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