Given that a = (-7, 20) and b = (3, -5), find the vector 6a + b.

July 20, 2026Mathematics

ScanSolve AI Answer

July 20, 2026

Answer

$\vec{a} = \begin{pmatrix} -\frac{15}{4} \\ 8 \end{pmatrix}, \quad \vec{b} = \begin{pmatrix} \frac{17}{4} \\ 5 \end{pmatrix}$

Step 1: Write down the given vector equations. We are given two vector equations: $\vec{a} - \vec{b} = \begin{pmatrix} -8 \\ 3 \end{pmatrix} \quad (1)$ $3\vec{a} + \vec{b} = \begin{pmatrix} -7 \\ 29 \end{pmatrix} \quad (2)$

Step 2: Add equation (1) and equation (2) to eliminate $\vec{b}$ . Adding the left-hand sides and right-hand sides of the two equations: $(\vec{a} - \vec{b}) + (3\vec{a} + \vec{b}) = \begin{pmatrix} -8 \\ 3 \end{pmatrix} + \begin{pmatrix} -7 \\ 29 \end{pmatrix}$ $4\vec{a} = \begin{pmatrix} -8 + (-7) \\ 3 + 29 \end{pmatrix}$ $4\vec{a} = \begin{pmatrix} -15 \\ 32 \end{pmatrix}$

Step 3: Solve for $\vec{a}$ . Divide both sides by 4: $\vec{a} = \frac{1}{4} \begin{pmatrix} -15 \\ 32 \end{pmatrix}$ $\vec{a} = \begin{pmatrix} \frac{-15}{4} \\ \frac{32}{4} \end{pmatrix}$ $\vec{a} = \begin{pmatrix} -\frac{15}{4} \\ 8 \end{pmatrix}$

Step 4: Substitute the value of $\vec{a}$ into equation (1) to find $\vec{b}$ . Using equation (1): $\vec{a} - \vec{b} = \begin{pmatrix} -8 \\ 3 \end{pmatrix}$ Substitute the value of $\vec{a}$ : $\begin{pmatrix} -\frac{15}{4} \\ 8 \end{pmatrix} - \vec{b} = \begin{pmatrix} -8 \\ 3 \end{pmatrix}$ Rearrange to solve for $\vec{b}$ : $\vec{b} = \begin{pmatrix} -\frac{15}{4} \\ 8 \end{pmatrix} - \begin{pmatrix} -8 \\ 3 \end{pmatrix}$ $\vec{b} = \begin{pmatrix} -\frac{15}{4} - (-8) \\ 8 - 3 \end{pmatrix}$ $\vec{b} = \begin{pmatrix} -\frac{15}{4} + 8 \\ 5 \end{pmatrix}$ To simplify the x-component, convert 8 to a fraction with denominator 4: $8 = \frac{32}{4}$ . $\vec{b} = \begin{pmatrix} -\frac{15}{4} + \frac{32}{4} \\ 5 \end{pmatrix}$ $\vec{b} = \begin{pmatrix} \frac{32 - 15}{4} \\ 5 \end{pmatrix}$ $\vec{b} = \begin{pmatrix} \frac{17}{4} \\ 5 \end{pmatrix}$

The vectors are: $\vec{a = \begin{pmatrix} -\frac{15}{4} \\ 8 \end{pmatrix}, \quad \vec{b} = \begin{pmatrix} \frac{17}{4} \\ 5 \end{pmatrix}}$

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Write down the given vector equations. We are given two vector equations: a - b = -8 \\ 3 (1) 3a + b = -7 \\ 29 (2) Step 2: Add equation (1) and equation (2) to eliminate b.

Asked on July 20, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on July 20, 2026

Step 1: Write down the given vector equations. We are given two vector equations: a - b = -8 \\ 3 (1) 3a + b = -7 \\ 29 (2) Step 2: Add equation (1) and equation (2) to eliminate b. Adding the left-hand sides and right-hand sides of the two equations: (a - b) + (3a + b) = -8 \\ 3 + -7 \\ 29 4a = -8 + (-7) \\ 3 + 29 4a = -15 \\ 32 Step 3: Solve for a. Divide both sides by 4: a = (1)/(4) -15 \\ 32 a = (-15)/(4) \\ (32)/(4) a = -(15)/(4) \\ 8 Step 4: Substitute the value of a into equation (1) to find b. Using equation (1): a - b = -8 \\ 3 Substitute the value of a: -(15)/(4) \\ 8 - b = -8 \\ 3 Rearrange to solve for b: b = -(15)/(4) \\ 8 - -8 \\ 3 b = -(15)/(4) - (-8) \\ 8 - 3 b = -(15)/(4) + 8 \\ 5 To simplify the x-component, convert 8 to a fraction with denominator 4: 8 = (32)/(4). b = -(15)/(4) + (32)/(4) \\ 5 b = (32 - 15)/(4) \\ 5 b = (17)/(4) \\ 5 The vectors are: a = -(15)/(4) \\ 8 , b = (17)/(4) \\ 5 Send me the next one 📸

Asked on July 20, 2026Mathematics

ScanSolve AI AnswerAnswered on July 20, 2026

Step 1: Write down the given vector equations. We are given two vector equations: a - b = -8 \\ 3 (1) 3a + b = -7 \\ 29 (2) Step 2: Add equation (1) and equation (2) to eliminate b. Adding the left-hand sides and right-hand sides of the two equations: (a - b) + (3a + b) = -8 \\ 3 + -7 \\ 29 4a = -8 + (-7) \\ 3 + 29 4a = -15 \\ 32 Step 3: Solve for a. Divide both sides by 4: a = (1)/(4) -15 \\ 32 a = (-15)/(4) \\ (32)/(4) a = -(15)/(4) \\ 8 Step 4: Substitute the value of a into equation (1) to find b. Using equation (1): a - b = -8 \\ 3 Substitute the value of a: -(15)/(4) \\ 8 - b = -8 \\ 3 Rearrange to solve for b: b = -(15)/(4) \\ 8 - -8 \\ 3 b = -(15)/(4) - (-8) \\ 8 - 3 b = -(15)/(4) + 8 \\ 5 To simplify the x-component, convert 8 to a fraction with denominator 4: 8 = (32)/(4). b = -(15)/(4) + (32)/(4) \\ 5 b = (32 - 15)/(4) \\ 5 b = (17)/(4) \\ 5 The vectors are: a = -(15)/(4) \\ 8 , b = (17)/(4) \\ 5 Send me the next one 📸