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Linear Equations: Practice Worksheet

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

18 problems·35–45 min·★★★★★
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  1. Problem 1
    Graphical Solution·★★★★★
    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
    xy
    ▶Answer
    The pair is consistent and has infinitely many solutions. There is no single point to report.
    ▶Step-by-step solution
    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
    xy
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  2. Problem 2
    Solving Linear Pairs·★★★★★
    Solve the equations $2x + \frac{2}{y} = -\frac{48}{5}$ and $5x + \frac{1}{y} = -\frac{124}{5}$, where y is non-zero.
    ▶Answer
    The solution is $(-5, 5)$.
    ▶Step-by-step solution
    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 = $5$.
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  3. Problem 3
    Word Problems·★★★★★
    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.
    ▶Answer
    The number is 62.
    ▶Step-by-step solution
    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.
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  4. Problem 4
    Word Problems·★★★★☆
    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.
    ▶Answer
    The present ages are 12 years and 36 years.
    ▶Step-by-step solution
    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 = $36$.
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  5. Problem 5
    Word Problems·★★★★★
    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.
    ▶Answer
    The fixed charge is Rs 12 and the extra charge is Rs 2 per day.
    ▶Step-by-step solution
    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 = $2$.
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  6. Problem 6
    Word Problems·★★★★★
    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.
    ▶Answer
    The stream speed is 6 km/h.
    ▶Step-by-step solution
    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 = $6$.
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  7. Problem 7
    Graphical Interpretation·★★☆☆☆
    Classify the two lines represented by $x + 3y = -4$ and $x + 3y = 2$.
    Sketch of the two lines
    xy
    ▶Answer
    The two lines are parallel.
    ▶Step-by-step solution
    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.
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  8. Problem 8
    Consistency·★★☆☆☆
    How many solutions does the pair of equations $2x + 3y = 6$ and $6x + 9y = 18$ have?
    ▶Answer
    The pair has infinitely many solutions.
    ▶Step-by-step solution
    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.
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  9. Problem 9
    Consistency·★★☆☆☆
    If a pair of linear equations in two variables is consistent, what can be said about the two lines?
    ▶Answer
    The lines are either intersecting or coincident.
    ▶Step-by-step solution
    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.
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  10. Problem 10
    Graphical Interpretation·★★★☆☆
    The equations $y = 5$ and $y = 1$ represent two horizontal lines. How many solutions do they have?
    ▶Answer
    They have no solution.
    ▶Step-by-step 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.
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  11. Problem 11
    Graphical Interpretation·★★★☆☆
    The equations $x = -3$ and $y = 2$ intersect at which point?
    ▶Answer
    They intersect at $(-3, 2)$.
    ▶Step-by-step solution
    1. The equation $x = -3$ fixes x at $-3$.
    2. The equation $y = 2$ fixes y at $2$.
    3. Therefore, the intersection point is $(-3, 2)$.
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  12. Problem 12
    Dependent Equations·★★★☆☆
    One equation of a dependent pair is $2x - 6y + 7 = 0$. Write another equation that could form the pair.
    ▶Answer
    One possible second equation is $-4x + 12y - 14 = 0$.
    ▶Step-by-step solution
    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.
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  13. Problem 13
    Constructing Systems·★★★☆☆
    Write a pair of linear equations that has the unique solution $(5, 4)$.
    ▶Answer
    One such pair is $5x - 3y = 13$ and $4x - 5y = 0$.
    ▶Step-by-step solution
    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.
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  14. Problem 14
    Solving Linear Pairs·★★★★☆
    Solve the pair of equations $-6x - y = 3$ and $4x + 5y = 11$.
    ▶Answer
    The solution is $(-1, 3)$.
    ▶Step-by-step solution
    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 $(-1, 3)$.
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  15. Problem 15
    Consistency Reasoning·★★★★☆
    Does the pair of equations $3x - 2y = -4$ and $3x - 2y = 4$ have no solution? Justify your answer.
    ▶Answer
    Yes. The pair is inconsistent and the lines are parallel.
    ▶Step-by-step solution
    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.
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  16. Problem 16
    Consistency Reasoning·★★★★☆
    Do the equations $-4x - 7y = 10$ and $12x + 21y = -30$ represent coincident lines? Justify your answer.
    ▶Answer
    Yes. The pair is consistent and the lines are coincident.
    ▶Step-by-step solution
    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.
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  17. Problem 17
    Consistency Reasoning·★★★★☆
    Are the equations $-7x - 4y = -21$ and $-x - 3y = -20$ consistent? Justify your answer.
    ▶Answer
    Yes. The pair is consistent and the lines are intersecting.
    ▶Step-by-step solution
    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.
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  18. Problem 18
    Graphical Interpretation·★★★★★
    Two straight paths are represented by $-5x + 4y = 11$ and $-5x + 4y = 3$. Do the paths cross each other?
    Sketch of the two lines
    xy
    ▶Answer
    No, the paths do not cross.
    ▶Step-by-step solution
    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.
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