Let \( \mathbf{u}=\langle 5,-2\rangle, \mathbf{v}=\langle-2,8\rangle \), and \( \mathbf{w}=\langle 0,-6\rangle \). Express \( 7 \mathbf{u}-2 \mathbf{v}+\mathbf{w} \) in the form \( \langle\mathrm{a}, \mathrm{b}\rangle \). \( 7 \mathbf{u}-2 \mathbf{v}+\mathbf{w}=\langle\square \) (Simplify your answers.)

Let's calculate \( 7 \mathbf{u} - 2 \mathbf{v} + \mathbf{w} \) step by step. First, we find \( 7 \mathbf{u} \): \[ 7 \mathbf{u} = 7 \langle 5, -2 \rangle = \langle 7 \cdot 5, 7 \cdot -2 \rangle = \langle 35, -14 \rangle \] Next, we calculate \( -2 \mathbf{v} \): \[ -2 \mathbf{v} = -2 \langle -2, 8 \rangle = \langle -2 \cdot -2, -2 \cdot 8 \rangle = \langle 4, -16 \rangle \] Now, we consider \( \mathbf{w} \): \[ \mathbf{w} = \langle 0, -6 \rangle \] Now we combine these results: \[ 7 \mathbf{u} - 2 \mathbf{v} + \mathbf{w} = \langle 35, -14 \rangle + \langle 4, -16 \rangle + \langle 0, -6 \rangle \] We add the components: \[ \text{For the x-components: } 35 + 4 + 0 = 39 \] \[ \text{For the y-components: } -14 - 16 - 6 = -36 \] Thus, the expression \( 7 \mathbf{u} - 2 \mathbf{v} + \mathbf{w} \) simplifies to: \[ \langle 39, -36 \rangle \] So, the final answer is: \[ 7 \mathbf{u} - 2 \mathbf{v} + \mathbf{w} = \langle 39, -36 \rangle \]