Systems of Linear Equations: math practice, Grade 9 – download the PDF

1 Is it a solution? ★★★

Consider the system \(y = 3x - 1\) and \(x + y = 7\). Decide whether each pair is a solution: (a) \((2, 5)\) (b) \((3, 4)\).

2 Table of values ★★★

Complete the table for the lines \(y = x + 2\) and \(y = -x + 6\) and use it to find the solution of the system.

3 A quick substitution ★★★

Solve the system \(y = 2x\) and \(x + y = 12\) by substitution.

4 A quick elimination ★★★

Solve the system \(x + y = 15\) and \(x - y = 3\) by elimination.

5 How many solutions? ★★★

Without solving, decide whether each system has one solution, no solution, or infinitely many. Compare the slopes and intercepts.

1. \(y = 4x + 1\) and \(y = 4x - 5\)
2. \(y = -x + 2\) and \(y = x + 2\)
3. \(2x + 2y = 6\) and \(y = -x + 3\)

6 Chess club ★★★

A chess club has 31 members. There are 9 more boys than girls. Write a system and find how many boys and how many girls are in the club.

7 Test points in a region ★★★

Consider the system \(y \ge 2x\) and \(y \lt x + 3\). For each point, say whether it is a solution of the system: \((1, 2)\), \((2, 3)\), \((0, 0)\).

8 Substitution with negatives ★★★

Solve by substitution: \(y = 3x - 9\) and \(5x + 2y = 4\).

9 Isolate first ★★★

Solve the system \(x + 3y = 11\) and \(2x - 5y = 0\). Start by isolating \(x\) in the first equation.

10 Elimination, one multiplier ★★★

Solve by elimination: \(4x + 3y = 25\) and \(2x - y = 5\).

11 Elimination, two multipliers ★★★

Solve by elimination: \(3x + 4y = 1\) and \(5x - 6y = 27\).

12 Reading a graph ★★★

The graph shows the two lines \(y = 0.5x + 1\) (line 1) and \(y = -x + 7\) (line 2).

(a) Estimate the coordinates of the point \(P\) from the graph. (b) Confirm your answer by solving the system algebraically.

13 Pool tickets ★★★

At a community pool, 4 adult tickets and 3 child tickets cost 41 dollars. Two adult tickets and 5 child tickets cost 31 dollars. Find the price of each kind of ticket.

14 A parameter ★★★

Consider the system \(y = 3x + 2\) and \(y = kx - 5\), where \(k\) is a number.

1. For which value of \(k\) does the system have no solution?
2. Solve the system when \(k = 2\).

15 Finding the missing constant ★★★

Consider the system \(2x + 3y = 12\) and \(4x + 6y = c\).

1. Find the value of \(c\) for which the system has infinitely many solutions.
2. Explain why the system has no solution when \(c = 20\).

16 Kayak trip ★★★

A kayaker paddles 12 miles (about 19.3 km) downstream in 2 hours, then 12 miles back upstream in 3 hours. Let \(b\) be the kayaker’s speed in still water and \(c\) the speed of the current, both in miles per hour. Find \(b\) and \(c\).

17 Juice mixture ★★★

A juice shop mixes a drink that is 15% real juice with a drink that is 40% real juice to make 10 gallons (about 37.9 liters) of a mixture that is 25% real juice. How many gallons of each drink are needed?

18 Fractions as coefficients ★★★

Solve the system \(\dfrac{1}{2}x + \dfrac{1}{3}y = 4\) and \(x - y = 3\). Start by clearing the fractions.

19 Ages ★★★

Mia is three times as old as her brother Leo. In 6 years, Mia will be twice as old as Leo will be then. How old are they now?

20 Shopping constraints ★★★

A student can spend at most 30 dollars on notebooks (2 dollars each) and pens (1 dollar each) and wants at least 5 items in all. Let \(n\) be the number of notebooks and \(p\) the number of pens, so \(n \ge 0\) and \(p \ge 0\).

1. Write a system of inequalities for the situation.
2. Is buying 8 notebooks and 10 pens possible? What about 12 notebooks and 8 pens?
3. What is the greatest number of items the student can buy?

21 A triangular region ★★★

Consider the region described by \(y \ge x - 2\), \(y \le -x + 4\) and \(y \ge 0\).

1. Find the three corner points of the region.
2. Find its area.