Practice Worksheet

Graphical and algebraic solutions, consistency, and applications of linear pairs.

Name
Class
Date

Answer all questions.

Show all necessary working.

Use a separate notebook for your solutions.

Use the graph of the pair $-7x + 8y = -30$ and $-14x + 16y = -60$ to decide whether the equations are consistent. If they are consistent, give the solution set.
Line pair graph
Solve the equations $2x + \frac{2}{y} = -\frac{48}{5}$ and $5x + \frac{1}{y} = -\frac{124}{5}$, where y is non-zero.
A two-digit code equals 9 times the sum of its digits minus 10. It also equals 14 times the difference of its digits plus 6. Find the code.
4 years ago, an older relative was 3 times as old as a younger relative. 8 years from now, the older relative will be 4 years more than twice the younger relative's age. Find their present ages.
A reading club charges one fixed amount for the first 2 days and then a daily extra amount. One reader paid Rs 22 for 7 days, and another paid Rs 16 for 4 days. Find the fixed charge and the extra charge per day.
A rower moves at 12 km/h in still water. For the same 36 km distance, the upstream trip takes 3 times the downstream trip. Find the speed of the stream.
Classify the two lines represented by $x + 3y = -4$ and $x + 3y = 2$.
Sketch of the two lines
How many solutions does the pair of equations $2x + 3y = 6$ and $6x + 9y = 18$ have?
If a pair of linear equations in two variables is consistent, what can be said about the two lines?
The equations $y = 5$ and $y = 1$ represent two horizontal lines. How many solutions do they have?
The equations $x = -3$ and $y = 2$ intersect at which point?
One equation of a dependent pair is $2x - 6y + 7 = 0$. Write another equation that could form the pair.
Write a pair of linear equations that has the unique solution $(5, 4)$.
Solve the pair of equations $-6x - y = 3$ and $4x + 5y = 11$.
Does the pair of equations $3x - 2y = -4$ and $3x - 2y = 4$ have no solution? Justify your answer.
Do the equations $-4x - 7y = 10$ and $12x + 21y = -30$ represent coincident lines? Justify your answer.
Are the equations $-7x - 4y = -21$ and $-x - 3y = -20$ consistent? Justify your answer.
Two straight paths are represented by $-5x + 4y = 11$ and $-5x + 4y = 3$. Do the paths cross each other?
Sketch of the two lines

Neokaal Study

Answer Key

The pair is consistent and has infinitely many solutions. There is no single point to report.
1. The pair is coincident.
2. Parallel distinct lines are inconsistent; coincident lines are consistent with infinitely many common points.
3. So the pair is consistent.
Solved line pair graph
The solution is $(-5, 5)$.
1. Let $u = \frac{1}{y}$.
2. Then the equations become $10x + 10u = -48$ and $25x + 5u = -124$ in x and u.
3. Solving gives x = $-5$ and u = $\frac{1}{5}$.
4. So y = $5$.
The number is 62.
1. Let x be the tens digit and y be the units digit.
2. The first condition gives $-x + 8y = 10$.
3. The second condition gives $4x - 15y = -6$.
4. Solving gives x = $6$ and y = $2$, so the number is 62.
The present ages are 12 years and 36 years.
1. Let x be the younger age and y be the older age.
2. The past condition gives $-3x + y = -8$.
3. The future condition gives $-2x + y = 12$.
4. Solving gives x = $12$ and y = $36$.
The fixed charge is Rs 12 and the extra charge is Rs 2 per day.
1. Let x be the fixed charge and y be the extra charge per day.
2. The two bills give $x + 5y = 22$ and $x + 2y = 16$.
3. Solving gives x = $12$ and y = $2$.
The stream speed is 6 km/h.
1. Upstream speed is $12 - v$ and downstream speed is $12 + v$.
2. The upstream time is 3 times the downstream time.
3. So $\frac{36}{12 - v} = 3\times \frac{36}{12 + v}$.
4. Solving gives v = $6$.
The two lines are parallel.
1. Compare the coefficient pairs of $x + 3y = -4$ and $x + 3y = 2$.
2. The determinant is $0$.
3. The x and y coefficients are proportional, but the constants are not in the same ratio.
4. So the lines are parallel and the pair has no solution.
5. Therefore, the pair has no solution.
The pair has infinitely many solutions.
1. Compare the coefficient pairs of $2x + 3y = 6$ and $6x + 9y = 18$.
2. The determinant is $0$.
3. All coefficients and constants are in the same ratio.
4. So the lines are coincident and every point on the line is a solution.
5. Therefore, the pair has infinitely many solutions.
The lines are either intersecting or coincident.
1. A consistent pair has at least one solution.
2. Intersecting lines have one common point, so they give one solution.
3. Coincident lines have every point common, so they give infinitely many solutions.
4. Parallel distinct lines have no common point, so they are not consistent.
They have no solution.
1. The first line is horizontal at y = $5$.
2. The second line is horizontal at y = $1$.
3. Distinct horizontal lines are parallel.
4. Parallel distinct lines do not meet, so the pair has no solution.
They intersect at $(-3, 2)$.
1. The equation $x = -3$ fixes x at $-3$.
2. The equation $y = 2$ fixes y at $2$.
3. Therefore, the intersection point is $(-3, 2)$.
One possible second equation is $-4x + 12y - 14 = 0$.
1. Dependent equations represent the same line.
2. Multiplying every term of an equation by the same non-zero number gives an equivalent equation.
3. Multiplying the given equation by $-2$ gives $-4x + 12y - 14 = 0$.
4. So the two equations form a dependent pair.
One such pair is $5x - 3y = 13$ and $4x - 5y = 0$.
1. Choose two different lines passing through $(5, 4)$.
2. The equation $5x - 3y = 13$ is true at that point.
3. The equation $4x - 5y = 0$ is also true at that point.
4. Their determinant is non-zero, so they meet only at that one point.
The solution is $(-1, 3)$.
1. Eliminate one variable from $-6x - y = 3$ and $4x + 5y = 11$.
2. The determinant is $-26$, so the pair has a unique solution.
3. Solving gives x = $-1$ and y = $3$.
4. Therefore, the solution is $(-1, 3)$.
Yes. The pair is inconsistent and the lines are parallel.
1. Compare the coefficient pairs of $3x - 2y = -4$ and $3x - 2y = 4$.
2. The determinant is $0$.
3. The x and y coefficients are proportional, but the constants are not in the same ratio.
4. So the lines are parallel and the pair has no solution.
5. Therefore, the pair has no solution.
Yes. The pair is consistent and the lines are coincident.
1. Compare the coefficient pairs of $-4x - 7y = 10$ and $12x + 21y = -30$.
2. The determinant is $0$.
3. All coefficients and constants are in the same ratio.
4. So the lines are coincident and every point on the line is a solution.
5. Therefore, the pair has infinitely many solutions.
Yes. The pair is consistent and the lines are intersecting.
1. Compare the coefficient pairs of $-7x - 4y = -21$ and $-x - 3y = -20$.
2. The determinant is $17$.
3. Since the determinant is non-zero, the lines intersect at one point.
4. The solution is $(-1, 7)$.
5. Therefore, the pair has one solution.
No, the paths do not cross.
1. Compare the coefficient pairs of $-5x + 4y = 11$ and $-5x + 4y = 3$.
2. The determinant is $0$.
3. The x and y coefficients are proportional, but the constants are not in the same ratio.
4. So the lines are parallel and the pair has no solution.
5. Therefore, the pair has no solution.