Solve y=x+1 and x²+y=7.

Question

ScanSolve AI Tutor · Accepted Answer

Question 10: Solve y = x + 1 and x² + y = 7. Step 1: Substitute the first equation into the second equation. x² + (x + 1) = 7 Step 2: Simplify and rearrange the equation into a standard quadratic form. x² + x + 1 − 7 = 0; x² + x − 6 = 0 Step 3: Factor the quadratic equation. (x + 3)(x − 2) = 0; This gives two possible values for x: x = −3 or x = 2. Step 4: Substitute these x values back into the equation y = x + 1 to find the corresponding y values. For x = −3: y = −3 + 1 = −2 So, one solution is (−3, −2). For x = 2: y = 2 + 1 = 3 So, the other solution is (2, 3). The solutions are (2, 3) and (−3, −2). The correct option is C. C. (x, y) = (2, 3) or (−3, −2); Question 11: y varies directly as x². If y = 18 when x = 3, find y when x = 5. Step 1: Write the direct variation equation. y = kx² where k is the constant of proportionality. Step 2: Use the given values (y = 18 when x = 3) to find k. 18 = k(3²); 18 = 9k; k = 18/9; k = 2; Step 3: Use the value of k to find y when x = 5. y = 2(5²); y = 2(25); y = 50 The correct option is C. C. 50 Question 12: y varies inversely as x. If y = 4 when x = 6, find y when x = 8. Step 1: Write the inverse variation equation. y = k/x where k is the constant of proportionality. Step 2: Use the given values (y = 4 when x = 6) to find k. 4 = k/6; k = 4 × 6; k = 24; Step 3: Use the value of k to find y when x = 8. y = 24/8; y = 3 The correct option is A. A. 3 Question 13: Find the 12th term of the arithmetic progression 5, 8, 11, ... Step 1: Identify the first term (a₁) and the common difference (d). The first term a₁ = 5. The common difference d = 8 − 5 = 3. Step 2: Use the formula for the n-th term of an arithmetic progression, a_n = a₁ + (n−1)d, to find the 12th term (n=12). a₁₂ = 5 + (12 − 1) × 3; a₁₂ = 5 + (11) × 3; a₁₂ = 5 + 33; a₁₂ = 38 The correct option is A. A. 38 Question 14: Find the sum of the first 10 terms of the arithmetic progression 2, 5, 8, ... Step 1: Identify the first term (a₁), the common difference (d), and the number of terms (n). The first term a₁ = 2. The common difference d = 5 − 2 = 3. The number of terms n = 10. Step 2: Use the formula for the sum of the first n terms of an arithmetic progression, S_n = n/2(2a₁ + (n−1)d). S₁₀ = 10/2(2(2) + (10 − 1)3); S₁₀ = 5(4 + (9)3); S₁₀ = 5(4 + 27); S₁₀ = 5(31); S₁₀ = 155 The correct option is A. A. 155 Question 15: Find the 5th term of the geometric progression 3, 6, 12, ... Step 1: Identify the first term (a₁) and the common ratio (r). The first term a₁ = 3. The common ratio r = 6/3 = 2. Step 2: Use the formula for the n-th term of a geometric progression, a_n = a₁ rⁿ⁻¹, to find the 5th term (n=5). a₅ = 3 × (2)⁵⁻¹; a₅ = 3 × (2)⁴; a₅ = 3 × 16; a₅ = 48 The correct option is A. A. 48 Question 16: Find the sum of the first 4 terms of the geometric progression 2, 6, 18, ... Step 1: Identify the first term (a₁), the common ratio (r), and the number of terms (n). The first term a₁ = 2. The common ratio r = 6/2 = 3. The number of terms n = 4. Step 2: Use the formula for the sum of the first n terms of a geometric progression, S_n = (a₁(rⁿ − 1))/(r − 1). S₄ = (2(3⁴ − 1))/(3 − 1); S₄ = (2(81 − 1))/(2); S₄ = (2(80))/(2); S₄ = 80 The correct option is A. A. 80 Question 17: Find the gradient of the line through (2, 3) and (6, 11). Step 1: Label the coordinates of the two points. Let (x₁, y₁) = (2, 3) and (x₂, y₂) = (6, 11). Step 2: Use the formula for the gradient (slope) of a line, m = (y₂ − y₁)/(x₂ − x₁). m = (11 − 3)/(6 − 2); m = 8/4; m = 2 The correct option is C. C. 2 Question 18: Find the distance between (−1, 2) and (3, 5). Step 1: Label the coordinates of the two points. Let (x₁, y₁) = (−1, 2) and (x₂, y₂) = (3, 5). Step 2: Use the distance formula, d = √((x₂ − x₁)² + (y₂ − y₁)²). d = √((3 − (−1))² + (5 − 2)²); d = √((3 + 1)² + (3)²); d = √((4)² + 9); d = √(16 + 9); d = √(25); d = 5 The distance is 5 units. The correct option is D. D. 5 units Question 19: Find the equation of the line with gradient 3 passing through (1, 2). Step 1: Use the point-slope form of a linear equation, y − y₁ = m(x − x₁). Given gradient m = 3 and point (x₁, y₁) = (1, 2). Step 2: Substitute the values into the formula. y − 2 = 3(x − 1); Step 3: Simplify the equation to the slope-intercept form (y = mx + c). y − 2 = 3x − 3; y = 3x − 3 + 2; y = 3x − 1 The correct option is B. B. y = 3x − 1 Question 20: Evaluate sin 30° + cos 60°. Step 1: Recall the standard trigonometric values for sin 30° and cos 60°. sin 30° = 1/2; cos 60° = 1/2 Step 2: Add the values. sin 30° + cos 60° = 1/2 + 1/2; sin 30° + cos 60° = 1 The correct option is D. D. 1

Point	x	y
x-intercept	−1	0
y-intercept	0	1

Solve y=x+1 and x²+y=7.

Answer

Step-by-step solution

Part 10· 4 steps

Substitute the first equation into the second equation.

Simplify and rearrange the equation into a standard quadratic form.

Factor the quadratic equation.

Substitute these xxx values back into the equation y=x+1y = x + 1y=x+1 to find the corresponding yyy values.

Part 11· 3 steps

Write the direct variation equation.

Use the given values (y=18y = 18y=18 when x=3x = 3x=3) to find kkk.

Use the value of kkk to find yyy when x=5x = 5x=5.

Part 12· 3 steps

Write the inverse variation equation.

Use the given values (y=4y = 4y=4 when x=6x = 6x=6) to find kkk.

Use the value of kkk to find yyy when x=8x = 8x=8.

Part 13· 2 steps

Identify the first term (a1a_1a1​) and the common difference (ddd).

Use the formula for the nnn-th term of an arithmetic progression, an=a1+(n−1)da_n = a_1 + (n-1)dan​=a1​+(n−1)d, to find the 12th term (n=12n=12n=12).

Part 14· 2 steps

Identify the first term (a1a_1a1​), the common difference (ddd), and the number of terms (nnn).

Use the formula for the sum of the first nnn terms of an arithmetic progression, Sn=n2(2a1+(n−1)d)S_n = \frac{n}{2}(2a_1 + (n-1)d)Sn​=2n​(2a1​+(n−1)d).

Part 15· 2 steps

Identify the first term (a1a_1a1​) and the common ratio (rrr).

Use the formula for the nnn-th term of a geometric progression, an=a1rn−1a_n = a_1 r^{n-1}an​=a1​rn−1, to find the 5th term (n=5n=5n=5).

Part 16· 2 steps

Identify the first term (a1a_1a1​), the common ratio (rrr), and the number of terms (nnn).

Use the formula for the sum of the first nnn terms of a geometric progression, Sn=a1(rn−1)r−1S_n = \frac{a_1(r^n - 1)}{r - 1}Sn​=r−1a1​(rn−1)​.

Part 17· 2 steps

Label the coordinates of the two points.

Use the formula for the gradient (slope) of a line, m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}m=x2​−x1​y2​−y1​​.

Part 18· 2 steps

Label the coordinates of the two points.

Use the distance formula, d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2​−x1​)2+(y2​−y1​)2​.

Part 19· 3 steps

Use the point-slope form of a linear equation, y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​).

Substitute the values into the formula.

Simplify the equation to the slope-intercept form (y=mx+cy = mx + cy=mx+c).

Part 20· 2 steps

Recall the standard trigonometric values for sin⁡30∘\sin 30^\circsin30∘ and cos⁡60∘\cos 60^\circcos60∘.

Add the values.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Substitute these $x$ values back into the equation $y = x + 1$ to find the corresponding $y$ values.

Use the given values ( $y = 18$ when $x = 3$ ) to find $k$ .

Use the value of $k$ to find $y$ when $x = 5$ .

Use the given values ( $y = 4$ when $x = 6$ ) to find $k$ .

Use the value of $k$ to find $y$ when $x = 8$ .

Identify the first term ( $a_1$ ) and the common difference ( $d$ ).

Use the formula for the $n$ -th term of an arithmetic progression, $a_n = a_1 + (n-1)d$ , to find the 12th term ( $n=12$ ).

Identify the first term ( $a_1$ ), the common difference ( $d$ ), and the number of terms ( $n$ ).

Use the formula for the sum of the first $n$ terms of an arithmetic progression, $S_n = \frac{n}{2}(2a_1 + (n-1)d)$ .

Identify the first term ( $a_1$ ) and the common ratio ( $r$ ).

Use the formula for the $n$ -th term of a geometric progression, $a_n = a_1 r^{n-1}$ , to find the 5th term ( $n=5$ ).

Identify the first term ( $a_1$ ), the common ratio ( $r$ ), and the number of terms ( $n$ ).

Use the formula for the sum of the first $n$ terms of a geometric progression, $S_n = \frac{a_1(r^n - 1)}{r - 1}$ .

Use the formula for the gradient (slope) of a line, $m = \frac{y_2 - y_1}{x_2 - x_1}$ .

Use the distance formula, $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

Use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$ .

Simplify the equation to the slope-intercept form ( $y = mx + c$ ).

Recall the standard trigonometric values for $\sin 30^\circ$ and $\cos 60^\circ$ .