Question

The system of equations above is graphed in the xy -plane. At which of the following points do the graphs of the equations intersect?

- ( - 3, - 2 )
- ( - 3, 2 )
- ( 5, - 2 )
- (5, 14 )

Hint:

**Hint:**

The given two equations are graphed on the xy plane. Obviously, every point on the curve will satisfy the equation. The point at which the lines intersect will satisfy both the given equations. Such a point will be the solution of the system of equations. So, we simply need to solve the given system of equations to get the point which intersects the graphs of the equations.

## The correct answer is: (5, 14 )

### The given equations are:

We will use the method of substitution to solve these equations.

From the first equation, we use the value of in the second equation,

We get an equation in one variable.

Simplifying this, we get

Taking all the terms on one side, we have

Thus, we get a quadratic equation,

factorization to solve We use middle term the above equation.

factorization to solve We use middle term the above equation.

Now, taking common from the first two terms and 2 from the last two terms, we have

So, we get two solutions for given by

Using these values of x in the first equation, we get two values of y

Thus, we get two solutions for the given system of equation

We can see that ( 5, 14 ) is in the options.

The correct option is D).

**Note:**

We can solve the equations by other methods too, but the method of substitution is most suitable here. Also, instead of solving

by middle term factorisation, we could also use the quadratic formula, which is given by

Where the standard form of equation is

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**Note:**

We do not have the exact inequalities in the option, so we choose the one from the options which is the closest approximation of the inequalities that we have calculated. There are a couple of formulas and concepts used here, such as the equation of a line passing through two points and the concept that the region below a line is given by replacing the equality sign with less than or equal to in the standard form of the equation and vice versa.

During mineral formation, the same chemical compound can be come different minerals depending on the temperature and pressure at the time of formation. A phase diagram is a graph that shows the conditions that are needed to form each mineral. The graph above is a portion of the phase diagram for aluminosilicates, with the temperature t , in degrees

Celsius , on the horizontal axis, and the pressure p , in gigapascals (G Pa), on the

Which of the following systems of inequalities best describes the region where sillimanite can form?

**Note:**

We do not have the exact inequalities in the option, so we choose the one from the options which is the closest approximation of the inequalities that we have calculated. There are a couple of formulas and concepts used here, such as the equation of a line passing through two points and the concept that the region below a line is given by replacing the equality sign with less than or equal to in the standard form of the equation and vice versa.