Given that three, 𝑦 plus 𝑥 multiplied by three, five minus zero, seven is equal to the zero vector, find the values of 𝑥 and 𝑦.
We recall that the zero vector in two dimensions has 𝑥- and 𝑦-coordinate equal to zero. This means that we need to solve the vector equation three, 𝑦 plus 𝑥 multiplied by three, five minus zero, seven is equal to zero, zero. We begin by multiplying the vector three, five by the scalar 𝑥. 𝑥 multiplied by three is equal to three 𝑥, and 𝑥 multiplied by five is equal to five 𝑥.
Our next step is to recall that when adding or subtracting vectors, we simply add or subtract the corresponding components. Equating the 𝑥-components, we have three plus three 𝑥 minus zero is equal to zero. Subtracting three from both sides of this equation gives us three 𝑥 is equal to negative three. We can then divide both sides of this equation by three, giving us a value of 𝑥 equal to negative one.
Equating the 𝑦-components, we have 𝑦 plus five 𝑥 minus seven is equal to zero. We can add seven and subtract five 𝑥 from both sides of this equation so that 𝑦 is equal to seven minus five 𝑥. As 𝑥 is equal to negative one, this leaves us with 𝑦 is equal to seven minus five multiplied by negative one. This in turn simplifies to 𝑦 is equal to seven plus five. Adding seven and five gives us 12. The values of 𝑥 and 𝑦 that satisfy the equation are negative one and 12, respectively.