MPM2D · Unit 1 test review

Linear Systems: Test Review

Everything for the Unit 1 test on one page: the key facts from all 4 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Solving Linear Systems by Graphing

The solution of a linear system is the ordered pair $(x, y)$ that makes every equation true; on a graph it is the point of intersection of the lines.
To check a possible solution, substitute it into each original equation; it must work in all of them.
Different slopes: the lines cross once, so the system has exactly one solution.
Same slope, different y-intercepts: parallel lines, no solution. Same slope and same y-intercept: the same line, infinitely many solutions.
Standard form $A x + B y = C$ : find the intercepts (set $x = 0$ , then $y = 0$ ), or rearrange to $y = m x + b$ when $B \neq = 0$ .

Solving by Substitution

Substitution: isolate a variable in one equation, substitute into the other equation, solve, back-substitute, check.
Isolate a variable with coefficient $1$ or $- 1$ to avoid fractions.
Put the substituted expression in brackets: $2 x - 3 y = - 9$ with $y = 14 - 3 x$ becomes $2 x - 3 (14 - 3 x) = - 9$ .
If the variables cancel and leave a true statement (such as $4 = 4$ ), there are infinitely many solutions; a false statement (such as $12 = 5$ ) means no solution.
The solution is an ordered pair $(x, y)$ : finding one variable is only halfway.

Solving by Elimination

Opposite coefficients ( $+ 3 y$ and $- 3 y$ ): add the equations. Equal coefficients ( $2 b$ and $2 b$ ): subtract them.
Multiplying every term of an equation by the same non-zero number gives an equivalent equation (the same line).
To match coefficients, multiply to their lowest common multiple, e.g. $3 x$ and $2 x$ both become $6 x$ .
If every variable cancels: a true statement ( $0 = 0$ ) means infinitely many solutions; a false statement ( $0 = 3$ ) means no solution.
Choose elimination for two standard-form equations, substitution when a variable is already isolated, graphing for a picture or estimate.

Problems Using Linear Systems

Define each variable precisely, with its units, before writing equations.
Most problems give an amount equation (totals of quantities) and a value equation (totals of cost, value or interest).
Simple interest: $I = P r t$ ; for one year $I = P r$ , with the rate as a decimal ( $4% = 0.04$ ).
With a current $c$ and still-water speed $b$ : downstream speed $b + c$ , upstream speed $b - c$ ; distance = speed × time.
Verify the solution in the original words and state it in a sentence with units.

Where marks get lost

Checking a point in only one equation. $(2, 35)$ makes $C = 10 h + 15$ true, but it fails $C = 5 h + 35$ , so it is not the solution of the system. (Solving Linear Systems by Graphing)
Treating the constant in standard form as the y-intercept. For $2 x + 3 y = 12$ the y-intercept is $4$ (set $x = 0$ : $3 y = 12$ ), not $12$ . (Solving Linear Systems by Graphing)
Dividing only some terms by a negative when rearranging. $2 x - y = 4$ becomes $- y = - 2 x + 4$ , and dividing every term by $- 1$ gives $y = 2 x - 4$ , not $y = - 2 x + 4$ . (Solving Linear Systems by Graphing)
Mixing up the two special cases: parallel lines (same slope, different intercepts) have no solution; the same line written twice has infinitely many. (Solving Linear Systems by Graphing)
Reporting a point read off a hand-drawn graph, such as $(0.7, 1.7)$ , as exact. Check it in both equations; if it does not work exactly, it is only an estimate. (Solving Linear Systems by Graphing)
Substituting the isolated expression back into the same equation it came from. You get something like $14 = 14$ and wrongly conclude there are infinitely many solutions. It must go into the other equation. (Solving by Substitution)
Dropping the brackets: replacing $y$ in $2 x - 3 y = - 9$ by $14 - 3 x$ and writing $2 x - 42 - 3 x$ instead of $2 x - 3 (14 - 3 x) = 2 x - 42 + 9 x$ . (Solving by Substitution)
Isolating with a sign error: from $4 x - y = 7$ , writing $y = 7 - 4 x$ . Subtracting $4 x$ gives $- y = 7 - 4 x$ , and multiplying every term by $- 1$ gives $y = 4 x - 7$ . (Solving by Substitution)
Stopping after one variable. " $x = 3$ " is not the solution of a system; back-substitute to find $y$ and give the ordered pair $(3, 5)$ . (Solving by Substitution)
Reading $0 = 0$ as "the solution is $(0, 0)$ ". A true statement with no variables left means infinitely many solutions. (Solving by Substitution)
Multiplying only one side, or only some terms: $3 x + 2 y = 16$ times 2 is $6 x + 4 y = 32$ , not $6 x + 4 y = 16$ . (Solving by Elimination)
Sign errors when subtracting: $(6 x + 8 y) - (6 x - 9 y)$ gives $17 y$ , not $- y$ , because subtracting $- 9 y$ adds $9 y$ . (Solving by Elimination)
Adding when the coefficients are equal (or subtracting when they are opposite). $2 b + 2 b = 4 b$ does not eliminate anything; equal coefficients need subtraction. (Solving by Elimination)
Finding one variable and stopping. Elimination gives $x$ (or $y$ ); back-substitute for the other and state the ordered pair. (Solving by Elimination)
Rounding decimals like $\frac{1}{3} \approx 0.33$ instead of clearing fractions first, which produces an answer that does not check exactly. (Solving by Elimination)
Defining variables vaguely (" $x$ = adults"). Say exactly what is counted and in what unit: " $x$ = the number of adult tickets sold". (Problems Using Linear Systems)
Converting percents wrongly: $4%$ is $0.04$ (not $4$ or $0.4$ ), and $3.5%$ is $0.035$ (not $0.35$ ). (Problems Using Linear Systems)
Mixing up the two equations in a mixture problem, for example writing $6 p + 15 c = 9$ instead of $6 p + 15 c = 9 (30)$ : the value of the mix is its price per kilogram times its total mass. (Problems Using Linear Systems)
Using $b + c$ for both directions in a current problem. Against the current the speeds subtract: $b - c$ . (Problems Using Linear Systems)
Stopping at the values of x and y, or not checking that they make sense (whole numbers of tickets, positive amounts) and answering the actual question in a sentence. (Problems Using Linear Systems)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Solving Linear Systems by Graphing
1
Which ordered pair is the solution of the system $y = x + 2$ and $y = - 2 x + 8$ ?
Solving Linear Systems by Graphing
2
The lines $y = 3 x - 4$ and $y = - x + 8$ cross at one point. What is the y-coordinate of that point?
Solving Linear Systems by Graphing
3
What is the point of intersection of the vertical line $x = 3$ and the horizontal line $y = - 2$ ? Write it as an ordered pair.
Solving Linear Systems by Graphing
4
Put the steps for solving a linear system by graphing in order.
1. 1Graph both lines on the same set of axes
2. 2Check the point in both original equations, then state the solution
3. 3Read the coordinates of the point where the lines cross
4. 4Rewrite each equation in the form y = mx + b, or find its intercepts
Solving Linear Systems by Graphing
5
Two gym memberships are graphed with $x$ = number of visits and $y$ = total cost in dollars. The lines cross at $(8, 120)$ . What does this point tell you?
Solving Linear Systems by Graphing
6
On a hand-drawn graph, the lines $y = x + 1$ and $y = - 2 x + 3$ seem to cross at $(0.7, 1.7)$ . Which statement is correct?
Solving by Substitution
7
In which equation is it easiest to isolate a variable without creating fractions?
Solving by Substitution
8
Solve the system $x = 2 y - 1$ and $3 x - 4 y = 5$ . What is the value of $y$ ?
Solving by Substitution
9
You correctly substitute an expression from the first equation into the second equation, and all the variables cancel, leaving $7 = 7$ . What does this tell you?
Solving by Substitution
10
Isolate $y$ in the equation $5 x - y = 8$ . Write your answer starting with "y =".
Solving by Substitution
11
A pizzeria sold 84 pizzas on Friday. It sold 3 times as many large pizzas as small ones. How many large pizzas did it sell?
pizzas
Solving by Substitution
12
After substituting correctly and simplifying, match each result to what it means for the system.
x = 2
5 = 5
0 = −3
Solving by Elimination
13
Solve $2 x + y = 9$ and $3 x - y = 11$ by elimination. What is the value of $x$ ?
Solving by Elimination
14
Finish solving $4 x + 3 y = 7$ and $5 x - 2 y = 26$ . What is the value of $y$ ?
Solving by Elimination
15
Subtract the second equation from the first: $4 x - 3 y = 10$ and $x - 3 y = 1$ . Which equation do you get?
Solving by Elimination
16
While solving a system by elimination, every variable cancels and you are left with $0 = 8$ . What does this mean?
Solving by Elimination
17
Solve $2 x + 3 y = 13$ and $3 x + 2 y = 12$ . What is the value of $x$ ?
Solving by Elimination
18
Match each system to the method that suits it best and the reason.
y = 4x − 3 and 2x + 5y = 18
5x + 2y = 9 and 3x − 2y = 7
Estimate where y = 2x + 1 and y = −x + 4 cross, and see the picture
Problems Using Linear Systems
19
For the Robotics Club problem ($5000 split between 4% and 3.5%), which is the best definition of the variables?
Problems Using Linear Systems
20
A jar holds 45 coins, all dimes and quarters, worth $7.65 in total. Let $d$ be the number of dimes and $q$ the number of quarters. Which system models the situation?
Problems Using Linear Systems
21
How many litres of a 10% salt solution must be mixed with a 40% salt solution to make 30 L of a 20% salt solution?
L
Problems Using Linear Systems
22
Gym A charges a $50 sign-up fee plus $8 per visit. Gym B has no sign-up fee and charges $13 per visit. After how many visits is the total cost the same?
visits
Problems Using Linear Systems
23
The length of a rectangle is 3 m more than twice its width. Its perimeter is 36 m. What is its length?
m
Problems Using Linear Systems
24
Put the steps for solving a word problem with a linear system in order.
1. 1Write two equations, one for each fact in the problem
2. 2Verify the values in the original words and state the answer in a sentence
3. 3Solve the system by substitution, elimination or graphing
4. 4Define two variables, with units

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…