How to do system of equations

1 answer:

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Answer: The three methods most commonly used to solve systems of equation are substitution, elimination and augmented matrices.

Step-by-step explanation:

Substitution is a method of solving systems of equations by removing all but one of the variables in one of the equations and then solving that equation.

Elimination is another way to solve systems of equations by rewriting one of the equations in terms of only one variable. The elimination method achieves this by adding or subtracting equations from each other in order to cancel out one of the variables.

Augmented matrices can also be used to solve systems of equations. The augmented matrix consists of rows for each equation, columns for each variable, and an augmented column that contains the constant term on the other side of the equation.

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Using a directrix of y = −2 and a focus of (1, 6), what quadratic function is created? Question 3 options:

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If your directrix is a "y=" line, that means that the parabola opens either upwards or downwards (as opposed to the left or the right). Because it is in the character of a parabola to "hug" the focus, our parabola opens upwards. The vertex of a parabola sits exactly halfway between the directrix and the focus. Since our directrix is at y = -2 and the focus is at (1, 6) AND the parabola opens upward, the vertex is going to sit on the main transversal, which is also the "line" the focus sits on. The focus is on the line x = 1, so the vertex will also have that x coordinate. Halfway between the y points of the directrix and the focus, -2 and 6, respectively, is the y value of 2. So the vertex sits at (1, 2). The formula for this type of parabola is $(x-h)^2=4p(y-k)$ where h and k are the coordinates of the vertex and p is the DISTANCE that the focus is from the vertex. Our focus is 4 units from the vertex, so p = 4. Filling in our h, k, and p: $(x-1)^2=4(4)(y-2)$ . Simplifying a bit gives us $(x-1)^2=16(y-2)$ . We can begin to isolate the y by dividing both sides by 16 to get $\frac{1}{16}(x-1)^2=y-2$ . Then we can add 2 to both sides to get the final equation $f(x)=\frac{1}{16}(x-1)^2+2$ , choice 4 from above.