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A point (4, 5) is mapped by the rule (x, y) -> (3x + 2y, 5x + 4y). What are the coordinates of the image point?

Mathematics

ScanSolve AI Answer

Answer

(22, 40)

Step-by-step solution

5 steps

  1. Identify the given point and the mapping rule.

    The given point is (x,y)=(4,5)(x, y) = (4, 5).
    The mapping rule is given by:

    (xy)→(3x+2y5x+4y)\begin{pmatrix} x \\ y \end{pmatrix} \rightarrow \begin{pmatrix} 3x + 2y \\ 5x + 4y \end{pmatrix}

  2. Substitute the values of xx and yy into the mapping rule.

    Substitute x=4x=4 and y=5y=5 into the expressions for the image coordinates.
    The first component of the image is 3x+2y3x + 2y.
    The second component of the image is 5x+4y5x + 4y.

  3. Calculate the value of the first component.

    3x+2y=3(4)+2(5)3x + 2y = 3(4) + 2(5) =12+10= 12 + 10 =22= 22

  4. Calculate the value of the second component.

    5x+4y=5(4)+4(5)5x + 4y = 5(4) + 4(5) =20+20= 20 + 20 =40= 40

  5. Write the image as a coordinate pair.

    The image of $(4, 5)$ under the given mapping is $(22, 40)$.

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