Question

How do you work on a system of equations with the solution (4, – 3)?

Answer 1

Hint: We are given a point (4, – 3), we are asked to find the different equations for which (4, – 3) acts as a solution. To do so we will first learn what the solution of the equation means. After that, we will find the equation which holds true at the point (4, – 3). We are having only one solution. So, we will form a linear equation only.

Complete step by step answer:
We are asked to create the system of the equation which has the solution (4, – 3) means we have to create the equation whose solution is (4, – 3). For the system of equations, the solution is referred to as that term where all equations simultaneously give the same value. If we can make a different type of system of equations whose solution is (4, – 3), we can have linear or quadratic or cubic or many more. But since we are given only one solution, so let the system of the equation have just one solution and hence our equation is linear. We know that the linear equation is given as y = mx + c where m is the slope and c is the y-intercept. Now, we will consider the slope of our choice, and then using point (4, – 3) as (x, y) we will find the value of c, and using them we will form our equation of the line.
We can use the point (4, – 3) because it is given that (4, – 3) must be the solution of our system of equations. So, it means it must satisfy our equation. Now, for the first equation, let us consider that slope is 2. So, we have m = 2. Now, as we have point (4, – 3), so we have x = 4 and y = – 3. Using them in y = mx + c, we get,
\[\Rightarrow -3=2\times 4+c\]
So, we get,
\[\Rightarrow -3=8+c\]
Subtracting 8 both the sides, we get,
\[\Rightarrow -3-8=8-8+c\]
So, we get,
\[\Rightarrow -11=c\]
Hence, our equation is
\[y=2x-11......\left( i \right)\]
Now for the second equation, let us consider the slope as 1, so m = 1. Now, using (x, y) = (4, – 3) and m = 1 in y = mx + c. We get,
\[\Rightarrow -3=4\times 1+c\]
\[\Rightarrow -3=4+c\]
Subtracting 4 on both the sides, we get,
\[\Rightarrow -3-4=4-4+c\]
So, we get,
\[\Rightarrow c=-7\]
So, using m = 1 and c = – 7, we get our second equation as
\[y=x-7.......\left( ii \right)\]
So, we get our system of the equation as y = 2x – 11 and y = x – 7.

Note:
 It is not necessary that we can find only two equations. We can find any number of equations as we know from a point, the infinite line can pass. So, in our system of the equation having a solution as (4, – 3), we can have an infinite number of equations. Also, remember while simplifying that 2 = c is the same as c = 2. Some may make a mistake like 2 = c implies c = – 2.