# How do you solve y=2x+9 and y=7x+10 using substitution?

Apr 30, 2016

I found:
$x = - \frac{1}{5}$
$y = \frac{43}{5}$

#### Explanation:

Take the first equation and substitute for $y$ into the second:
$\textcolor{red}{2 x + 9} = 7 x + 10$
$5 x = - 1$
$x = - \frac{1}{5}$
now we substitute this value back into the first:
$y = - 2 \left(\textcolor{b l u e}{\frac{1}{5}}\right) + 9$
$y = \frac{43}{5}$

May 1, 2016

Here is another way of thinking about what is given....
The answer is the same as given by the previous contributor.

#### Explanation:

Both equations are in the form $y = \ldots \ldots$

In other words we have been given two different ways of writing $y$

The two values for $y$ are the same, so $y = y$
Therefore the other parts of each equation must be equal to each other as well, leading to $7 x + 10 = 2 x + 9$

I tend to regard this method as 'equating', rather than substitution.

This now gives an equation with one variable and it can be solved.

Once a value for $x$ has been found, it can be substituted into each of the given equations to find $y$.

The second substitution acts as a check to ensure that the answers are correct.

Remember that from a graphical point of view, solving the equations of two straight lines simultaneously, gives the point of intersection of the two lines.

This concept is extremely useful and important.