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Slope equals the change in y divided by the change in x: (11 − 3)/(6 − 2) = 8/4 = 2. The line rises 8 units while moving 4 units right, so the slope is positive 2.
Add 7 to both sides to get 3x = 21, then divide both sides by 3 to get x = 7. Substituting 7 back gives 21 − 7 = 14, which checks.
Substitute 3 for x: f(3) = 2(3) + 5 = 6 + 5 = 11. Multiplication is performed before addition, so 2 times 3 is computed first.
A line written as y = mx + b has y-intercept b, the value of y when x = 0. Here b = −9, so the line crosses the y-axis at (0, −9).
Substitute x = 1: y = 3(1) − 2 = 1, so the point (1, 1) satisfies the equation. Each other point fails when its x-value is substituted.
Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Substituting m = −3 and b = 4 gives y = −3x + 4.
Subtract 2 from both sides to get x/4 = 3, then multiply both sides by 4 to get x = 12. Substituting gives 12/4 + 2 = 3 + 2 = 5.
An absolute value equation splits into two cases: x − 3 = 7 or x − 3 = −7. Solving these gives x = 10 and x = −4, and both check in the original equation.
Substitute y = x + 1 into the first equation: 2x + (x + 1) = 10, so 3x + 1 = 10 and 3x = 9, giving x = 3. Then y = 4, and the pair (3, 4) satisfies both equations.
Rewrite 2x + 3y = 12 as y = −(2/3)x + 4, so the given line has slope −2/3. Perpendicular slopes are negative reciprocals, and the negative reciprocal of −2/3 is 3/2.
Model the total cost as 25 + 15m = 130, where m is the number of months. Subtracting 25 gives 15m = 105, so m = 7.
Substitute 3 for every x: f(3) = 3² − 4(3) = 9 − 12 = −3. The square is taken before the subtraction, so the result is negative.
Subtract 5 from both sides to get −2x > 6, then divide both sides by −2 and reverse the inequality symbol, giving x < −3.
Solve for y: −2y = −4x + 8, so y = 2x − 4. In slope-intercept form the coefficient of x is the slope, which is 2.
Parallel lines share the same slope, so the new line has slope 3 and form y = 3x + b. Substituting (2, 7) gives 7 = 6 + b, so b = 1 and the equation is y = 3x + 1.
The x-intercept occurs where y = 0. Substituting gives 3x = 12, so x = 4 and the intercept is the point (4, 0).
A linear system has no solution when the lines are parallel but distinct. The second line has slope −3, and the first has slope −k/2; setting −k/2 = −3 gives k = 6, and the intercepts 3 and 5 differ, so the lines are indeed distinct.
The average rate of change is (f(3) − f(1))/(3 − 1). Since f(3) = 10 and f(1) = 2, this is (10 − 2)/2 = 4.
Let a be the number of adult tickets, so 200 − a is the number of child tickets. Then 8a + 5(200 − a) = 1300, which simplifies to 3a + 1000 = 1300, so a = 100.
Multiply both sides of A = (1/2)bh by 2 to get 2A = bh, then divide both sides by b to isolate h: h = 2A/b.
An absolute value less than or equal to a positive number becomes a compound inequality: −9 ≤ 2x − 5 ≤ 9. Adding 5 gives −4 ≤ 2x ≤ 14, and dividing by 2 gives −2 ≤ x ≤ 7.
The slope is (19 − 7)/(5 − 2) = 12/3 = 4. Using f(2) = 7 gives 7 = 4(2) + b, so b = −1, and because f(0) = b, the value is −1.
The acid amounts must balance: 0.20(10) + 0.50x = 0.35(10 + x). This gives 2 + 0.5x = 3.5 + 0.35x, so 0.15x = 1.5 and x = 10.
From x − y = 1 we get y = x − 1. Substituting into 2x + 3y = 12 gives 5x − 3 = 12, so x = 3 and y = 2. Therefore x² − y² = 9 − 4 = 5.
Slope equals the change in y divided by the change in x: (k − 4)/(3 − 1) = 1/2. So (k − 4)/2 = 1/2, giving k − 4 = 1 and k = 5.
Substitute −3 for every x in the rule: f(−3) = 5 − 2(−3). The product −2 · (−3) equals 6, so f(−3) = 5 + 6 = 11.
Distribute the 3 to get 3x − 6 = 15, then add 6 to both sides to obtain 3x = 21. Dividing both sides by 3 gives x = 7.
Slope is the change in y divided by the change in x between two points on the line. Here the change in y is 13 − 5 = 8 and the change in x is 6 − 2 = 4, so the slope is 8/4 = 2.
A point lies on the graph exactly when its coordinates make the equation true. Substituting x = 3 and y = 2 gives 2(3) + 3(2) = 6 + 6 = 12, so (3, 2) is on the line.
Add 7 to both sides to get 4x ≤ 16, then divide both sides by 4. Because 4 is positive, the inequality symbol keeps its direction, giving x ≤ 4.
Every point on a horizontal line has the same y-coordinate, so the change in y between any two points is 0. Slope equals change in y divided by change in x, which gives 0 divided by a nonzero number, namely 0.
"Twice a number" is 2n, and "three less than" that quantity means 3 is subtracted from it, giving 2n − 3. Setting this equal to 17 gives 2n − 3 = 17.
In a linear model of the form V = a + bt, the coefficient b is the rate of change of the quantity per unit of time. Because t is measured in minutes and the coefficient is −12, the volume drops 12 liters per minute.
Multiply every term by the common denominator 6 to clear the fractions: 2x − x = 24. Combining like terms gives x = 24, and checking shows 8 − 4 = 4.
Isolate the y-term by subtracting 4x from both sides, which gives 3y = 24 − 4x. Then divide every term by 3 to get y = (24 − 4x)/3.
Adding the two equations eliminates y and gives 6x = 12, so x = 2. Substituting into 4x + 3y = 5 gives 8 + 3y = 5, so y = −1, and therefore x + y = 2 + (−1) = 1.
Distributing the left side gives 3x + 12 = 3x + 12, so the variable terms cancel and the equation reduces to the true statement 12 = 12. Since the statement holds for every value of x, every real number is a solution.
Let n be the number of notebooks; the total cost is 6n + 8, which must be at most 50. Solving 6n + 8 ≤ 50 gives 6n ≤ 42, so n ≤ 7, and the greatest whole number of notebooks is 7.
Let s be the smaller number; then the larger is 5s, and s + 5s = 42. Solving 6s = 42 gives s = 7, so the larger number is 5(7) = 35.
Replace every x in the rule with the entire expression a + 1: f(a + 1) = 4(a + 1) − 7. Distributing gives 4a + 4 − 7, which simplifies to 4a − 3.
Substitute x = −2 and y = 4 into y = 3x + b to get 4 = 3(−2) + b, so 4 = −6 + b. Adding 6 to both sides gives b = 10.
The rate of change of f is its slope, 3. For g the rate of change is (18 − 2)/(4 − 0) = 4, and since 4 > 3, g has the greater rate of change.
Cross-multiplying gives 2(x + 2) = 3(x − 1), so 2x + 4 = 3x − 3. Subtracting 2x from both sides gives 4 = x − 3, so x = 7, and checking gives 9/3 = 6/2 = 3.
Represent the integers as n, n + 2, and n + 4, where n is even. Then 3n + 6 = 84, so 3n = 78 and n = 26, giving 26 + 28 + 30 = 84.
Set the model equal to 8,000: 24,000 − 3,200t = 8,000. Subtracting 8,000 from both sides and adding 3,200t gives 16,000 = 3,200t, so t = 5.
Multiply every term by the common denominator 6 to get 2(x − 2) − 3(x + 1) ≥ 6. This simplifies to 2x − 4 − 3x − 3 ≥ 6, or −x − 7 ≥ 6, so −x ≥ 13 and therefore x ≤ −13 after dividing by −1 and reversing the symbol.
Split into two cases. If 2x − 1 = x + 4, then x = 5. If 2x − 1 = −(x + 4), then 2x − 1 = −x − 4, so 3x = −3 and x = −1. Both values make the right side x + 4 positive, so both are genuine solutions.
Multiplying every term by 10 clears the decimals: 2(5x + 10) = 5x + 80. Distributing gives 10x + 20 = 5x + 80, so 5x = 60 and x = 12.
Solve the equation for y to read the slope: by = 6 − 3x, so y = 6/b − (3/b)x, meaning the slope is −3/b. Setting −3/b = 1/2 and cross-multiplying gives −6 = b.
The slope of the given line is (10 − 2)/(5 − 1) = 2, so the slope of the perpendicular line k is the negative reciprocal, −1/2. Using the point (4, 1) in y = −1/2 x + b gives 1 = −2 + b, so b = 3 and the equation is y = −1/2 x + 3.