11.10 Systems of Equations

Solve $\;y = 2x + 1, \quad y = 7\;$ for $(x, y)$.

Solve $\;x + y = 10, \quad x - y = 4\;$.

Is $(x, y) = (4, 1)$ a solution of $\;3x + 2y = 14\;$?

Two straight lines meet at the point $(3, 5)$ on a graph. State the solution of the corresponding system of equations.

Solve $\;\dfrac{x}{2} + y = 5, \quad y = 2\;$.

Solve $\;y = 2x + 1, \quad 3x + y = 16\;$.

A coffee costs $c$ and a tea costs $t$ francs. 2 coffees and 3 teas cost 15 francs. Write an equation in $c$ and $t$.

Two equations $2x + 3y = 12$ and $4x + 6y = 24$ are graphed. State the number of solutions and explain.

Two equations $2x + 3y = 12$ and $2x + 3y = 18$ are graphed. State the number of solutions and explain.

For $\;y = x^2, \; y = x + 6\;$, what type of system is this?

Solve $\;2x + 3y = 12, \quad x - y = 1\;$.

A coffee costs $x$ francs and a tea costs $y$ francs. 3 coffees + 2 teas = 19; 2 coffees + 4 teas = 22. Find $x$ and $y$.

Solve $\;y = 3x - 4, \quad 2x + y = 11\;$.

A taxi charges a fixed fee plus a rate per km. A 5 km ride costs CHF 12 and an 8 km ride costs CHF 18. Find the fixed fee and the rate per km.

The system $\;x + 2y = 7, \; 3x + ky = 5\;$ has solution $x = 1$. Find $k$ and $y$.

Solve $\;y = x^2, \; y = x + 6\;$.

The graphs of $y = x + 1$ and $y = -x + 5$ meet at a single point. State the point and verify.

Use your GDC to solve $\;2.4x + 3.7y = 10, \quad 1.5x - 0.9y = 4.5\;$. Give $x$ and $y$ to 3 s.f.

Solve $\;\dfrac{x}{2} + \dfrac{y}{3} = 5, \quad x + y = 12\;$.

A graph shows demand $D = -2p + 100$ and supply $S = 3p + 5$ where $p$ is price. Find the equilibrium price and quantity.

Solve $\;x + y + z = 6, \quad 2x - y + z = 3, \quad x + 2y - z = 2\;$.

For what $k$ does $\;2x + 3y = 12, \; 4x + 6y = k\;$ have (a) inf many solutions, (b) no solution?

Find the values of $k$ for which $\;y = kx, \; y = x^2 + 1\;$ has no real solutions.

A boat travels 12 km downstream and 12 km back upstream in 5 hours. Still-water speed $v$ km/h; current 2 km/h. Set up and solve.

A business has cost $C = 200 + 8n$ and revenue $R = 12n$ for $n$ items. Find the break-even quantity.

Use technology to solve $\;x^2 + y^2 = 25, \; y = x + 1\;$. Give answers to 3 s.f.

Find the value of $k$ for which $\;y = 2x + 1, \; y = kx - 3\;$ has no solution.

A person splits CHF 5000 between two investments: account A pays 3% per year and B pays 5%. The total annual interest is CHF 210. How much was placed in each?

A quadratic $y = ax^2 + bx + c$ passes through $(1, 4)$, $(2, 9)$, $(3, 18)$. Use a 3×3 system to find $a$, $b$, $c$.

Find the intersections of the curves $y = x^2 - 4$ and $y = -x^2 + 2x$.

Find all real solutions of $\;y = 2x + 1, \; y = x^2 - 2\;$.

A quadratic $f(x) = ax^2 + bx + c$ satisfies $f(0) = 3$, $f(1) = 6$, $f(2) = 13$. Find $a$, $b$, $c$ and $f(-1)$.

A theatre sells adult tickets for $a$ francs and children for $c$ francs. Monday: 80 adult + 30 child = CHF 1500. Tuesday: 60 adult + 50 child = CHF 1400. Find $a$ and $c$.

Solve $\;\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{5}{6}, \quad \dfrac{1}{x} - \dfrac{1}{y} = \dfrac{1}{6}\;$.

Quadratic regression on data through $(1, 5)$, $(2, 12)$, $(3, 23)$ gives $y = ax^2 + bx + c$. Set up and solve the system for $a, b, c$.

Two pipes fill a swimming pool. Pipe A alone takes 8 h; pipe B alone takes 12 h. How long do they take together?

Find the intersections of $\;xy = 6, \; y = x + 1\;$.

Five years ago, Anna was 3 times as old as Ben. In 5 years, she will be twice as old. Find their current ages.

For what value(s) of $k$ does the system $\;x + y + z = 3, \; x - y + 2z = k, \; 2x + 3y - z = 5\;$ have a unique solution?

A printer charges either Plan A (CHF 30 fixed + CHF 8 per page) or Plan B (CHF 50 fixed + CHF 6 per page). (a) Find the number of pages for which both plans cost the same. (b) Which is cheaper for 25 pages?