# Solution of a system of equations - one of them is linear and the other quadratic

πPractice solution of a system of equations, one of which is linear and the other quadratic.

## Solving a system of equations when one of them is linear and the other is quadratic

When we have a system of equations where one of the equations is linear and the other quadratic
we will use the substitution method:

We will isolate one variable from an equation, place in the second equation the value of the expression of the variable we have isolated, and in this way, we will obtain an equation with one variable. We will solve for $X$ or $Y$ and then place it in one of the original equations to find the complete point. The point we discover will be the point of intersection of the line with the parabola, and it will also be the solution of the system of equations.

## Test yourself on solution of a system of equations, one of which is linear and the other quadratic.!

Solve the following system of equations:

$$\begin{cases} \sqrt{x}-\sqrt{y}=\sqrt{\sqrt{61}-6} \\ xy=9 \end{cases}$$

### Given the following system of equations:$y=x^2+2x+7$,$2x+y=4$

Solution:
Let's isolate one of the variables from one of the equations:
$2x+y=4$
$y=-2x+4$

Now, let's substitute the value we isolated into the second equation. We will obtain:

$-2x+4=x^2+2x+7$

Let's combine like terms and transpose members. We will obtain:
$-X^2-4x-3=0$
We will solve it with the help of the quadratic formula and will obtain:
$X=-1 ,-3$
Now we will gradually place an $X$ we found into one of the equations to find the complete points: Let's start with $X=-1$

$2\times(-1)+y=4$
$-2+y=4$
$y=6$
Let's write this solution $(-1,6)$
let's move to the second $X$ we found $X=-3$ and we will obtain:
$2*(-3)+y=4$
$-6+y=4$
$y=10$ let's write this solution $(-3,10)$

These solutions are the points of intersection of the parabola with the axis.

If you are interested in this article, you might also be interested in the following articles:

The functions y=xΒ²

Family of parabolas y=xΒ²+c: Vertical shift

Family of parabolas y=(x-p)Β²

Family of parabolas y=(x-p)Β²+k (combination of horizontal and vertical shift)

Vertex form of the quadratic function

Factored form of the quadratic function

Completing the square in a quadratic equation

Standard form of the quadratic function

System of quadratic equations - Algebraic and graphical solution

In the blog of Tutorela you will find a variety of articles about mathematics.

## Examples and exercises with solutions of linear and quadratic equation systems

### Exercise #1

Solve the following system of equations:

$\begin{cases} \sqrt{x}-\sqrt{y}=\sqrt{\sqrt{61}-6} \\ xy=9 \end{cases}$

### Video Solution

$x=\frac{\sqrt{61}}{2}-2.5$

$y=\frac{\sqrt{61}}{2}+2.5$

or

$x=\frac{\sqrt{61}}{2}+2.5$

$y=\frac{\sqrt{61}}{2}-2.5$

### Exercise #2

Solve the following system of equations:

$\begin{cases} \sqrt{x}+\sqrt{y}=\sqrt{\sqrt{61}+6} \\ xy=9 \end{cases}$

### Video Solution

$x=\frac{\sqrt{61}}{2}-2.5$

$y=\frac{\sqrt{61}}{2}+2.5$

or

$x=\frac{\sqrt{61}}{2}+2.5$

$y=\frac{\sqrt{61}}{2}-2.5$

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