/Linear algebra

Write the answers to the following questions on your own paper. Show your working clearly.
task 1 – part a
Two Mathematics students Piripi and Riana who had just completed learning about linear
systems at school were experimenting with the equations. They wrote down two equations and
to investigate the systems further they included an unknown coefficient in one of the equations.
Equation 1: 3x + y – 2z = –
7
Equation 2: –
az = x + y + 6
To form the third equation they asked their friend Fatima to suggest a 4 digit number. Fatima
came up with the number 2219. The digits 2, 2, 1, 9 will be the coefficients of x, y, z and the
constant term in that order, given that the equation is in the form: ax + by + cz = d.
• Write down Equation 3 and solve the set of three equations when a = –
3.
• Find the values of a, for which the system is consistent. Give algebraic reasons for your
answer and interpret the solution geometrically.
Piripi and Riana wanted to experiment with the equations still further continuing to use a = –
3.
Investigate the solutions to the set of equations described in the two situations given below.
Give a geometric interpretation of the solution for each set of equations and describe the
solution for each set.
task 1 – part b
A new set of three equations is formed using the original Equation 1 and Equation 2 and the
third equation is formed by multiplying all the coefficients and the constant in Equation 1 by 3.
task 1 – part c
A new set of three equations is formed using the original Equation 1 and Equation 2 and the
third equation is formed by using Equation 1 with the constant changed from -7 to 12.
2 MXO3151 © te aho o te kura pounamu
task 2
Tyrone inherited $25,000 and invested part of it in a money market account, part in bonus bonds
and part in a mutual fund. After one year, he received a total of $1,620 in simple interest from
the three investments. The money market paid 6% annually, the bonds paid 7% annually, and
the mutual fund paid 8% annually. There was $6000 more invested in the bonds than the mutual
funds.
• Set up and solve a system of equations to find the amount Tyrone invested in each
category