Linear-Quadraric Systems of Equations - Examples, Exercises and Solutions

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.

Practice Linear-Quadraric Systems of Equations

Question 1

The following functions are graphed below:

$$ f(x)=x^2-6x+8 $$

$$ g(x)=4x-17 $$

For which values of x is
$$ f(x)<0 $$ true?

Question 2

Look at the graph below of the following functions:

$$ f(x)=x^2-6x+8 $$

$$ g(x)=-x+4 $$

For which values of x is
$$ g(x)>0 $$ true?

Question 3

Which formula describes graph 2?

Question 4

Which formula represents line 2 in the graph below?

Question 5

Which formula represents line 1 in the graph below?

Examples with solutions for Linear-Quadraric Systems of Equations

Exercise #1

The following functions are graphed below:

$f(x)=x^2-6x+8$

$g(x)=4x-17$

For which values of x is
$f(x)<0$ true?

Video Solution

Answer

$2 < x < 4$

Exercise #2

Look at the graph below of the following functions:

$f(x)=x^2-6x+8$

$g(x)=-x+4$

For which values of x is
$g(x)>0$ true?

Video Solution

Answer

$x<4$

Exercise #3

Which formula describes graph 2?

Video Solution

Answer

$y=4x-17$

Exercise #4

Which formula represents line 2 in the graph below?

Video Solution

Answer

$y=-x+4$

Exercise #5

Which formula represents line 1 in the graph below?

Video Solution

Answer

$y=x^2-6x+8$

Question 1

Which formula represents line 2 shown in the graph below?

Question 2

Choose the formula that represents line 1 in the graph below:

Question 3

The following function is graphed below:

$$ f(x)=x^2-6x+8 $$

$$ g(x)=-x+4 $$

For which values of x is
$$ f(x) > g(x) $$ true?

Question 4

Choose the formula that describes graph 1:

Question 5

The following function is graphed below:

$$ g(x)=-x+4 $$

For which values of x is

$$ f(x) < g(x) $$ true?

Exercise #6

Which formula represents line 2 shown in the graph below?

Video Solution

Answer

$y=-2x+5$

Exercise #7

Choose the formula that represents line 1 in the graph below:

Video Solution

Answer

$y=x^2-6x$

Exercise #8

The following function is graphed below:

$f(x)=x^2-6x+8$

$g(x)=-x+4$

For which values of x is
$f(x) > g(x)$ true?

Video Solution

Answer

$x < 1,4 < x$

Exercise #9

Choose the formula that describes graph 1:

Video Solution

Answer

$y=x^2-6x+8$

Exercise #10

The following function is graphed below:

$g(x)=-x+4$

For which values of x is

$f(x) < g(x)$ true?

Video Solution

Answer

$1 < x < 4$

Question 1

The following functions are graphed below:

$$ f(x)=x^2-6x+8 $$

$$ g(x)=4x-17 $$

For which values of x is
$$ f(x) < g(x) $$ true?

Question 2

The following functions are graphed below:

$$ f(x)=x^2-6x+8 $$

$$ g(x)=4x-17 $$

For which values of x is

$$ f(x)>0 $$ true?

Question 3

Solve the following system of equations:

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

Question 4

Solve the following system of equations:

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

Exercise #11

The following functions are graphed below:

$f(x)=x^2-6x+8$

$g(x)=4x-17$

For which values of x is
$f(x) < g(x)$ true?

Video Solution

Answer

$5 < x$

Exercise #12

The following functions are graphed below:

$f(x)=x^2-6x+8$

$g(x)=4x-17$

For which values of x is

$f(x)>0$ true?

Video Solution

Answer

$x < 2, 4 < x$

Exercise #13

Solve the following system of equations:

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

Video Solution

Answer

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

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

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

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

Exercise #14

Solve the following system of equations:

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

Video Solution

Answer

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

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

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

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

Topics learned in later sections