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Solve for y in the equation 2x + y = 10 when x = 3.
y = 4
y = 6
y = 10
y = 2
Substitute x = 3 into 2x + y = 10 giving 2(3) + y = 10, so 6 + y = 10 and y = 4. This direct substitution method is the simplest way to solve for y when x is known. For more practice on one-variable linear equations, see Khan Academy.
For the equation y = 3x - 5, what is the y-intercept?
(0, -5)
(5, 0)
-5
3
In y = mx + b, b represents the y-intercept coordinate at (0, b). Here b = -5, so the line crosses the y-axis at (0, -5). Understanding slope-intercept form is key to graphing lines. For details on intercepts, see Math is Fun.
Identify the slope of the line represented by 4x - 2y = 8.
-2
2
4
-4
Rearrange to y = mx + b: 4x - 2y = 8 ? -2y = -4x + 8 ? y = 2x - 4, giving slope m = 2. Converting to slope-intercept form reveals the slope directly. More on converting forms at Khan Academy.
What is the solution (x, y) for the system x + y = 7 and x - y = 1?
(5, 2)
(4, 3)
(3, 4)
(2, 5)
Adding the equations gives 2x = 8, so x = 4, then substitute into x + y = 7 to get y = 3. This method of adding or subtracting equations is called elimination. For a step-by-step guide, check Math is Fun.
Solve the system 2x + 3y = 12 and x - y = 2.
(3, 2)
(3.6, 1.6)
(4, 1)
(2, 4)
From x - y = 2, x = y + 2. Substitute into 2x + 3y = 12 to get 2(y+2)+3y = 12 ? 5y + 4 = 12 ? y = 1.6, then x = 3.6. This is the substitution method. For more examples, see Khan Academy.
Determine the number of solutions for the system 2x + 4y = 8 and x + 2y = 4.
Two solutions
Exactly one solution
No solution
Infinite solutions
The second equation is exactly half of the first, so both represent the same line and there are infinitely many solutions. When two equations are scalar multiples, they coincide. Learn more about coincident lines at Math is Fun.
Express y in terms of x for the equation 6x - 3y = 12.
y = 2x - 4
y = -3x + 12
y = -2x + 4
y = (2/3)x - 4
Rearrange: 6x - 3y = 12 ? -3y = -6x + 12 ? y = 2x - 4. Isolating y gives slope-intercept form y = mx + b. See more on isolating variables at MathBitsNotebook.
Find the intersection of y = (1/2)x + 1 and y = -x + 4.
(2, 3)
(3, 2.5)
(2, 2)
(1, 1.5)
Set (1/2)x + 1 = -x + 4 ? 1.5x = 3 ? x = 2, then y = 2. This is solving by equating two expressions for y. More on graph intersections at Khan Academy.
Solve the system 3x - 2y = 7 and 5x + 4y = 1.
(2, -1)
(15/11, -16/11)
(-1, 2)
(1, 1)
Multiply the first equation by 2: 6x - 4y = 14, then add to 5x + 4y = 1 to get 11x = 15 ? x = 15/11. Substitute back to find y = -16/11. For elimination techniques, visit Khan Academy.
Determine k such that the system kx + 2y = 6 and 3x + 6y = 15 has no solution.
0.5
3
1
-1
Lines are parallel (no solution) when slopes match and intercepts differ. Here slope1 = -k/2 and slope2 = -3/6 = -1/2, so k = 1 makes slopes equal but different intercepts. Learn about parallel lines at Math is Fun.
A line passes through (2, 3) and (5, 11). What is its equation in slope-intercept form?
y = (8/3)x - 7/3
y = (7/3)x - 8/3
y = (3/8)x + 7/3
y = (8/3)x + 1/3
Slope m = (11 - 3)/(5 - 2) = 8/3. Use point-slope: y - 3 = (8/3)(x - 2) ? y = (8/3)x - 7/3. This process combines slope calculation with point substitution. More examples at Khan Academy.
Given 2x + 3y = 12 and 4x + ky = 24, for which value of k are the lines perpendicular?
-8/3
3/2
-2/3
4/3
Line1 slope = -2/3, so perpendicular slope = 3/2. In 4x + ky = 24, slope = -4/k. Set -4/k = 3/2 ? k = -8/3. Perpendicular slopes are negative reciprocals. See Math is Fun.
For the system ax + by = c and dx + ey = f, what condition on a, b, d, e guarantees a unique solution?
ae - bd = 0
ad + be ? 0
ab + de ? 0
ae - bd ? 0
A unique solution exists when the determinant of the coefficient matrix is nonzero: |a b; d e| = ae ? bd ? 0. This is the basis of Cramer's Rule. For a formal explanation, see Cramer's Rule.
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Study Outcomes
Understand the relationship between x and y in the equation -
Develop a clear conceptual grasp of how changes in one variable affect the other and interpret solution pairs effectively.
Apply algebraic methods effectively -
Use substitution and elimination techniques to systematically solve for x and y in linear equations and reinforce problem-solving strategies.
Solve for x and y accurately -
Obtain precise values for variables in various equation forms and verify your solutions against equation requirements.
Interpret graphical solutions -
Translate equation solutions into graphical representations and analyze the significance of intersection points on coordinate grids.
Evaluate contextual problem scenarios -
Apply your skills to real-world and word problems, interpreting the solution of a linear equation within practical contexts.
Strengthen conceptual confidence -
Identify knowledge gaps through immediate feedback and reinforce understanding with targeted practice on linear equations.
Cheat Sheet
Slope-Intercept Form Mastery -
Mastering the slope - intercept form y = mx + b gives you a quick way to graph and interpret linear relationships on any algebra linear equations test. According to Khan Academy, the slope m represents rise over run and the y-intercept b shows where the line crosses the y-axis. For example, y = 2x + 3 rises 2 units for every 1 unit across, making sketching almost instantaneous.
Inverse Operations for Variable Isolation -
To improve your knowledge of x and y in the equation, practice isolating the variable using inverse operations like addition/subtraction and multiplication/division. Purplemath recommends the mnemonic "Undo, Simplify, Solve" to remember the sequence. For instance, to solve 3x + 5 = 20, subtract 5 then divide by 3 to find x = 5.
Graphical Solutions & Coordinate Interpretation -
Interpreting the solution of a linear equation visually helps reinforce how (x, y) pairs represent points on a line. MIT OpenCourseWare shows that plotting points like (2, 1) or (−1, 4) confirms they satisfy y = −½x + 3. This hands-on method also builds intuition for your linear equations practice quiz.
Solution Verification via Substitution -
Always check answers by plugging your x and y back into the original equation to ensure they satisfy it. According to Math Is Fun, this step catches small errors and boosts confidence before exams. For example, substitute x = 4 into 2x − y = 3, then solve for y to confirm your result.
Translating Word Problems into Linear Models -
Real-world scenarios often hide the equation structure, so practice turning descriptions into ax + by = c form. The National Council of Teachers of Mathematics suggests identifying knowns, unknowns, and relationships step-by-step. For instance, if a taxi charges a $2 flat fee plus $1.50 per mile, model cost C as C = 1.5m + 2 to ace a solve for x and y quiz.
