Algebra
Calculus
Trigonometry
Matrix
Question
Solve the system of equations
(x1,y1)=(705+3205,10−5+3205)(x2,y2)=(705−3205,−105+3205)
Evaluate
{5xy=13∣3x−y∣=∣4x−1∣
Solve the equation for x
More Steps
Evaluate
5xy=13
Evaluate
5yx=13
Divide both sides
5y5yx=5y13
Divide the numbers
x=5y13
{x=5y13∣3x−y∣=∣4x−1∣
Substitute the given value of x into the equation ∣3x−y∣=∣4x−1∣
3×5y13−y=4×5y13−1
Simplify
More Steps
Evaluate
3×5y13−y
Multiply the terms
More Steps
Multiply the terms
3×5y13
Multiply the terms
5y3×13
Multiply the terms
5y39
5y39−y
Subtract the terms
More Steps
Simplify
5y39−y
Reduce fractions to a common denominator
5y39−5yy×5y
Write all numerators above the common denominator
5y39−y×5y
Multiply the terms
5y39−5y2
5y39−5y2
Rewrite the expression
51×y39−5y2
Rewrite the expression
51y39−5y2
51y39−5y2=4×5y13−1
Simplify
More Steps
Evaluate
4×5y13−1
Multiply the terms
More Steps
Multiply the terms
4×5y13
Multiply the terms
5y4×13
Multiply the terms
5y52
5y52−1
Subtract the terms
More Steps
Evaluate
5y52−1
Reduce fractions to a common denominator
5y52−5y5y
Write all numerators above the common denominator
5y52−5y
5y52−5y
Rewrite the expression
51×y52−5y
Rewrite the expression
51y52−5y
51y39−5y2=51y52−5y
Move the expression to the left side
51y39−5y2−51y52−5y=0
Separate the equation into 4 possible cases
51×y39−5y2−51×y52−5y=0,y39−5y2≥0,y52−5y≥051×y39−5y2−51(−y52−5y)=0,y39−5y2≥0,y52−5y<051(−y39−5y2)−51×y52−5y=0,y39−5y2<0,y52−5y≥051(−y39−5y2)−51(−y52−5y)=0,y39−5y2<0,y52−5y<0
Solve the equation
More Steps
Evaluate
51×y39−5y2−51×y52−5y=0
Calculate
More Steps
Evaluate
51×y39−5y2−51×y52−5y
Rearrange the terms
5y39−5y2−51×y52−5y
Rearrange the terms
5y39−5y2−5y52−5y
Write all numerators above the common denominator
5y39−5y2−(52−5y)
Subtract the terms
5y−13−5y2+5y
5y−13−5y2+5y=0
Cross multiply
−13−5y2+5y=5y×0
Simplify the equation
−13−5y2+5y=0
Rewrite in standard form
−5y2+5y−13=0
Multiply both sides
5y2−5y+13=0
Substitute a=5,b=−5 and c=13 into the quadratic formula y=2a−b±b2−4ac
y=2×55±(−5)2−4×5×13
Simplify the expression
y=105±(−5)2−4×5×13
Simplify the expression
More Steps
Evaluate
(−5)2−4×5×13
Multiply the terms
(−5)2−260
Rewrite the expression
52−260
Evaluate the power
25−260
Subtract the numbers
−235
y=105±−235
The expression is undefined in the set of real numbers
y∈/R
y∈/R,y39−5y2≥0,y52−5y≥051×y39−5y2−51(−y52−5y)=0,y39−5y2≥0,y52−5y<051(−y39−5y2)−51×y52−5y=0,y39−5y2<0,y52−5y≥051(−y39−5y2)−51(−y52−5y)=0,y39−5y2<0,y52−5y<0
Solve the inequality
More Steps
Evaluate
y39−5y2≥0
Separate the inequality into 2 possible cases
{39−5y2≥0y>0∪{39−5y2≤0y<0
Solve the inequality
More Steps
Evaluate
39−5y2≥0
Rewrite the expression
−5y2≥−39
Change the signs on both sides of the inequality and flip the inequality sign
5y2≤39
Divide both sides
55y2≤539
Divide the numbers
y2≤539
Take the 2-th root on both sides of the inequality
y2≤539
Calculate
∣y∣≤5195
Separate the inequality into 2 possible cases
{y≤5195y≥−5195
Find the intersection
−5195≤y≤5195
{−5195≤y≤5195y>0∪{39−5y2≤0y<0
Solve the inequality
More Steps
Evaluate
39−5y2≤0
Rewrite the expression
−5y2≤−39
Change the signs on both sides of the inequality and flip the inequality sign
5y2≥39
Divide both sides
55y2≥539
Divide the numbers
y2≥539
Take the 2-th root on both sides of the inequality
y2≥539
Calculate
∣y∣≥5195
Separate the inequality into 2 possible cases
y≥5195∪y≤−5195
Find the union
y∈(−∞,−5195]∪[5195,+∞)
{−5195≤y≤5195y>0∪{y∈(−∞,−5195]∪[5195,+∞)y<0
Find the intersection
0<y≤5195∪{y∈(−∞,−5195]∪[5195,+∞)y<0
Find the intersection
0<y≤5195∪y≤−5195
Find the union
y∈(−∞,−5195]∪(0,5195]
y∈/R,y∈(−∞,−5195]∪(0,5195],y52−5y≥051×y39−5y2−51(−y52−5y)=0,y39−5y2≥0,y52−5y<051(−y39−5y2)−51×y52−5y=0,y39−5y2<0,y52−5y≥051(−y39−5y2)−51(−y52−5y)=0,y39−5y2<0,y52−5y<0
Solve the inequality
More Steps
Evaluate
y52−5y≥0
Separate the inequality into 2 possible cases
{52−5y≥0y>0∪{52−5y≤0y<0
Solve the inequality
More Steps
Evaluate
52−5y≥0
Move the constant to the right side
−5y≥0−52
Removing 0 doesn't change the value,so remove it from the expression
−5y≥−52
Change the signs on both sides of the inequality and flip the inequality sign
5y≤52
Divide both sides
55y≤552
Divide the numbers
y≤552
{y≤552y>0∪{52−5y≤0y<0
Solve the inequality
More Steps
Evaluate
52−5y≤0
Move the constant to the right side
−5y≤0−52
Removing 0 doesn't change the value,so remove it from the expression
−5y≤−52
Change the signs on both sides of the inequality and flip the inequality sign
5y≥52
Divide both sides
55y≥552
Divide the numbers
y≥552
{y≤552y>0∪{y≥552y<0
Find the intersection
0<y≤552∪{y≥552y<0
Find the intersection
0<y≤552∪y∈∅
Find the union
0<y≤552
y∈/R,y∈(−∞,−5195]∪(0,5195],0<y≤55251×y39−5y2−51(−y52−5y)=0,y39−5y2≥0,y52−5y<051(−y39−5y2)−51×y52−5y=0,y39−5y2<0,y52−5y≥051(−y39−5y2)−51(−y52−5y)=0,y39−5y2<0,y52−5y<0
Solve the equation
More Steps
Evaluate
51×y39−5y2−51(−y52−5y)=0
Calculate
More Steps
Evaluate
51×y39−5y2−51(−y52−5y)
Rearrange the terms
5y39−5y2−51(−y52−5y)
Rearrange the terms
5y39−5y2+5y52−5y
Write all numerators above the common denominator
5y39−5y2+52−5y
Add the numbers
5y91−5y2−5y
5y91−5y2−5y=0
Cross multiply
91−5y2−5y=5y×0
Simplify the equation
91−5y2−5y=0
Rewrite in standard form
−5y2−5y+91=0
Multiply both sides
5y2+5y−91=0
Substitute a=5,b=5 and c=−91 into the quadratic formula y=2a−b±b2−4ac
y=2×5−5±52−4×5(−91)
Simplify the expression
y=10−5±52−4×5(−91)
Simplify the expression
More Steps
Evaluate
52−4×5(−91)
Multiply
52−(−1820)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
52+1820
Evaluate the power
25+1820
Add the numbers
1845
y=10−5±1845
Simplify the radical expression
More Steps
Evaluate
1845
Write the expression as a product where the root of one of the factors can be evaluated
9×205
Write the number in exponential form with the base of 3
32×205
The root of a product is equal to the product of the roots of each factor
32×205
Reduce the index of the radical and exponent with 2
3205
y=10−5±3205
Separate the equation into 2 possible cases
y=10−5+3205y=10−5−3205
Use b−a=−ba=−ba to rewrite the fraction
y=10−5+3205y=−105+3205
Evaluate the logic
y=10−5+3205∪y=−105+3205
y∈/R,y∈(−∞,−5195]∪(0,5195],0<y≤552y=10−5+3205∪y=−105+3205,y39−5y2≥0,y52−5y<051(−y39−5y2)−51×y52−5y=0,y39−5y2<0,y52−5y≥051(−y39−5y2)−51(−y52−5y)=0,y39−5y2<0,y52−5y<0
Solve the inequality
More Steps
Evaluate
y39−5y2≥0
Separate the inequality into 2 possible cases
{39−5y2≥0y>0∪{39−5y2≤0y<0
Solve the inequality
More Steps
Evaluate
39−5y2≥0
Rewrite the expression
−5y2≥−39
Change the signs on both sides of the inequality and flip the inequality sign
5y2≤39
Divide both sides
55y2≤539
Divide the numbers
y2≤539
Take the 2-th root on both sides of the inequality
y2≤539
Calculate
∣y∣≤5195
Separate the inequality into 2 possible cases
{y≤5195y≥−5195
Find the intersection
−5195≤y≤5195
{−5195≤y≤5195y>0∪{39−5y2≤0y<0
Solve the inequality
More Steps
Evaluate
39−5y2≤0
Rewrite the expression
−5y2≤−39
Change the signs on both sides of the inequality and flip the inequality sign
5y2≥39
Divide both sides
55y2≥539
Divide the numbers
y2≥539
Take the 2-th root on both sides of the inequality
y2≥539
Calculate
∣y∣≥5195
Separate the inequality into 2 possible cases
y≥5195∪y≤−5195
Find the union
y∈(−∞,−5195]∪[5195,+∞)
{−5195≤y≤5195y>0∪{y∈(−∞,−5195]∪[5195,+∞)y<0
Find the intersection
0<y≤5195∪{y∈(−∞,−5195]∪[5195,+∞)y<0
Find the intersection
0<y≤5195∪y≤−5195
Find the union
y∈(−∞,−5195]∪(0,5195]
y∈/R,y∈(−∞,−5195]∪(0,5195],0<y≤552y=10−5+3205∪y=−105+3205,y∈(−∞,−5195]∪(0,5195],y52−5y<051(−y39−5y2)−51×y52−5y=0,y39−5y2<0,y52−5y≥051(−y39−5y2)−51(−y52−5y)=0,y39−5y2<0,y52−5y<0
Solve the inequality
More Steps
Evaluate
y52−5y<0
Separate the inequality into 2 possible cases
{52−5y>0y<0∪{52−5y<0y>0
Solve the inequality
More Steps
Evaluate
52−5y>0
Move the constant to the right side
−5y>0−52
Removing 0 doesn't change the value,so remove it from the expression
−5y>−52
Change the signs on both sides of the inequality and flip the inequality sign
5y<52
Divide both sides
55y<552
Divide the numbers
y<552
{y<552y<0∪{52−5y<0y>0
Solve the inequality
More Steps
Evaluate
52−5y<0
Move the constant to the right side
−5y<0−52
Removing 0 doesn't change the value,so remove it from the expression
−5y<−52
Change the signs on both sides of the inequality and flip the inequality sign
5y>52
Divide both sides
55y>552
Divide the numbers
y>552
{y<552y<0∪{y>552y>0
Find the intersection
y<0∪{y>552y>0
Find the intersection
y<0∪y>552
Find the union
y∈(−∞,0)∪(552,+∞)
y∈/R,y∈(−∞,−5195]∪(0,5195],0<y≤552y=10−5+3205∪y=−105+3205,y∈(−∞,−5195]∪(0,5195],y∈(−∞,0)∪(552,+∞)51(−y39−5y2)−51×y52−5y=0,y39−5y2<0,y52−5y≥051(−y39−5y2)−51(−y52−5y)=0,y39−5y2<0,y52−5y<0
Solve the equation
More Steps
Evaluate
51(−y39−5y2)−51×y52−5y=0
Calculate
More Steps
Evaluate
51(−y39−5y2)−51×y52−5y
Rearrange the terms
−5y39−5y2−51×y52−5y
Rearrange the terms
−5y39−5y2−5y52−5y
Write all numerators above the common denominator
5y−(39−5y2)−(52−5y)
Subtract the terms
5y−91+5y2+5y
5y−91+5y2+5y=0
Cross multiply
−91+5y2+5y=5y×0
Simplify the equation
−91+5y2+5y=0
Rewrite in standard form
5y2+5y−91=0
Substitute a=5,b=5 and c=−91 into the quadratic formula y=2a−b±b2−4ac
y=2×5−5±52−4×5(−91)
Simplify the expression
y=10−5±52−4×5(−91)
Simplify the expression
More Steps
Evaluate
52−4×5(−91)
Multiply
52−(−1820)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
52+1820
Evaluate the power
25+1820
Add the numbers
1845
y=10−5±1845
Simplify the radical expression
More Steps
Evaluate
1845
Write the expression as a product where the root of one of the factors can be evaluated
9×205
Write the number in exponential form with the base of 3
32×205
The root of a product is equal to the product of the roots of each factor
32×205
Reduce the index of the radical and exponent with 2
3205
y=10−5±3205
Separate the equation into 2 possible cases
y=10−5+3205y=10−5−3205
Use b−a=−ba=−ba to rewrite the fraction
y=10−5+3205y=−105+3205
Evaluate the logic
y=10−5+3205∪y=−105+3205
y∈/R,y∈(−∞,−5195]∪(0,5195],0<y≤552y=10−5+3205∪y=−105+3205,y∈(−∞,−5195]∪(0,5195],y∈(−∞,0)∪(552,+∞)y=10−5+3205∪y=−105+3205,y39−5y2<0,y52−5y≥051(−y39−5y2)−51(−y52−5y)=0,y39−5y2<0,y52−5y<0
Solve the inequality
More Steps
Evaluate
y39−5y2<0
Separate the inequality into 2 possible cases
{39−5y2>0y<0∪{39−5y2<0y>0
Solve the inequality
More Steps
Evaluate
39−5y2>0
Rewrite the expression
−5y2>−39
Change the signs on both sides of the inequality and flip the inequality sign
5y2<39
Divide both sides
55y2<539
Divide the numbers
y2<539
Take the 2-th root on both sides of the inequality
y2<539
Calculate
∣y∣<5195
Separate the inequality into 2 possible cases
{y<5195y>−5195
Find the intersection
−5195<y<5195
{−5195<y<5195y<0∪{39−5y2<0y>0
Solve the inequality
More Steps
Evaluate
39−5y2<0
Rewrite the expression
−5y2<−39
Change the signs on both sides of the inequality and flip the inequality sign
5y2>39
Divide both sides
55y2>539
Divide the numbers
y2>539
Take the 2-th root on both sides of the inequality
y2>539
Calculate
∣y∣>5195
Separate the inequality into 2 possible cases
y>5195∪y<−5195
Find the union
y∈(−∞,−5195)∪(5195,+∞)
{−5195<y<5195y<0∪{y∈(−∞,−5195)∪(5195,+∞)y>0
Find the intersection
−5195<y<0∪{y∈(−∞,−5195)∪(5195,+∞)y>0
Find the intersection
−5195<y<0∪y>5195
Find the union
y∈(−5195,0)∪(5195,+∞)
y∈/R,y∈(−∞,−5195]∪(0,5195],0<y≤552y=10−5+3205∪y=−105+3205,y∈(−∞,−5195]∪(0,5195],y∈(−∞,0)∪(552,+∞)y=10−5+3205∪y=−105+3205,y∈(−5195,0)∪(5195,+∞),y52−5y≥051(−y39−5y2)−51(−y52−5y)=0,y39−5y2<0,y52−5y<0
Solve the inequality
More Steps
Evaluate
y52−5y≥0
Separate the inequality into 2 possible cases
{52−5y≥0y>0∪{52−5y≤0y<0
Solve the inequality
More Steps
Evaluate
52−5y≥0
Move the constant to the right side
−5y≥0−52
Removing 0 doesn't change the value,so remove it from the expression
−5y≥−52
Change the signs on both sides of the inequality and flip the inequality sign
5y≤52
Divide both sides
55y≤552
Divide the numbers
y≤552
{y≤552y>0∪{52−5y≤0y<0
Solve the inequality
More Steps
Evaluate
52−5y≤0
Move the constant to the right side
−5y≤0−52
Removing 0 doesn't change the value,so remove it from the expression
−5y≤−52
Change the signs on both sides of the inequality and flip the inequality sign
5y≥52
Divide both sides
55y≥552
Divide the numbers
y≥552
{y≤552y>0∪{y≥552y<0
Find the intersection
0<y≤552∪{y≥552y<0
Find the intersection
0<y≤552∪y∈∅
Find the union
0<y≤552
y∈/R,y∈(−∞,−5195]∪(0,5195],0<y≤552y=10−5+3205∪y=−105+3205,y∈(−∞,−5195]∪(0,5195],y∈(−∞,0)∪(552,+∞)y=10−5+3205∪y=−105+3205,y∈(−5195,0)∪(5195,+∞),0<y≤55251(−y39−5y2)−51(−y52−5y)=0,y39−5y2<0,y52−5y<0
Solve the equation
More Steps
Evaluate
51(−y39−5y2)−51(−y52−5y)=0
Calculate
More Steps
Evaluate
51(−y39−5y2)−51(−y52−5y)
Rearrange the terms
−5y39−5y2−51(−y52−5y)
Rearrange the terms
−5y39−5y2+5y52−5y
Write all numerators above the common denominator
5y−(39−5y2)+52−5y
Calculate the sum or difference
5y13+5y2−5y
5y13+5y2−5y=0
Cross multiply
13+5y2−5y=5y×0
Simplify the equation
13+5y2−5y=0
Rewrite in standard form
5y2−5y+13=0
Substitute a=5,b=−5 and c=13 into the quadratic formula y=2a−b±b2−4ac
y=2×55±(−5)2−4×5×13
Simplify the expression
y=105±(−5)2−4×5×13
Simplify the expression
More Steps
Evaluate
(−5)2−4×5×13
Multiply the terms
(−5)2−260
Rewrite the expression
52−260
Evaluate the power
25−260
Subtract the numbers
−235
y=105±−235
The expression is undefined in the set of real numbers
y∈/R
y∈/R,y∈(−∞,−5195]∪(0,5195],0<y≤552y=10−5+3205∪y=−105+3205,y∈(−∞,−5195]∪(0,5195],y∈(−∞,0)∪(552,+∞)y=10−5+3205∪y=−105+3205,y∈(−5195,0)∪(5195,+∞),0<y≤552y∈/R,y39−5y2<0,y52−5y<0
Solve the inequality
More Steps
Evaluate
y39−5y2<0
Separate the inequality into 2 possible cases
{39−5y2>0y<0∪{39−5y2<0y>0
Solve the inequality
More Steps
Evaluate
39−5y2>0
Rewrite the expression
−5y2>−39
Change the signs on both sides of the inequality and flip the inequality sign
5y2<39
Divide both sides
55y2<539
Divide the numbers
y2<539
Take the 2-th root on both sides of the inequality
y2<539
Calculate
∣y∣<5195
Separate the inequality into 2 possible cases
{y<5195y>−5195
Find the intersection
−5195<y<5195
{−5195<y<5195y<0∪{39−5y2<0y>0
Solve the inequality
More Steps
Evaluate
39−5y2<0
Rewrite the expression
−5y2<−39
Change the signs on both sides of the inequality and flip the inequality sign
5y2>39
Divide both sides
55y2>539
Divide the numbers
y2>539
Take the 2-th root on both sides of the inequality
y2>539
Calculate
∣y∣>5195
Separate the inequality into 2 possible cases
y>5195∪y<−5195
Find the union
y∈(−∞,−5195)∪(5195,+∞)
{−5195<y<5195y<0∪{y∈(−∞,−5195)∪(5195,+∞)y>0
Find the intersection
−5195<y<0∪{y∈(−∞,−5195)∪(5195,+∞)y>0
Find the intersection
−5195<y<0∪y>5195
Find the union
y∈(−5195,0)∪(5195,+∞)
y∈/R,y∈(−∞,−5195]∪(0,5195],0<y≤552y=10−5+3205∪y=−105+3205,y∈(−∞,−5195]∪(0,5195],y∈(−∞,0)∪(552,+∞)y=10−5+3205∪y=−105+3205,y∈(−5195,0)∪(5195,+∞),0<y≤552y∈/R,y∈(−5195,0)∪(5195,+∞),y52−5y<0
Solve the inequality
More Steps
Evaluate
y52−5y<0
Separate the inequality into 2 possible cases
{52−5y>0y<0∪{52−5y<0y>0
Solve the inequality
More Steps
Evaluate
52−5y>0
Move the constant to the right side
−5y>0−52
Removing 0 doesn't change the value,so remove it from the expression
−5y>−52
Change the signs on both sides of the inequality and flip the inequality sign
5y<52
Divide both sides
55y<552
Divide the numbers
y<552
{y<552y<0∪{52−5y<0y>0
Solve the inequality
More Steps
Evaluate
52−5y<0
Move the constant to the right side
−5y<0−52
Removing 0 doesn't change the value,so remove it from the expression
−5y<−52
Change the signs on both sides of the inequality and flip the inequality sign
5y>52
Divide both sides
55y>552
Divide the numbers
y>552
{y<552y<0∪{y>552y>0
Find the intersection
y<0∪{y>552y>0
Find the intersection
y<0∪y>552
Find the union
y∈(−∞,0)∪(552,+∞)
y∈/R,y∈(−∞,−5195]∪(0,5195],0<y≤552y=10−5+3205∪y=−105+3205,y∈(−∞,−5195]∪(0,5195],y∈(−∞,0)∪(552,+∞)y=10−5+3205∪y=−105+3205,y∈(−5195,0)∪(5195,+∞),0<y≤552y∈/R,y∈(−5195,0)∪(5195,+∞),y∈(−∞,0)∪(552,+∞)
Find the intersection
y∈/Ry=10−5+3205∪y=−105+3205,y∈(−∞,−5195]∪(0,5195],y∈(−∞,0)∪(552,+∞)y=10−5+3205∪y=−105+3205,y∈(−5195,0)∪(5195,+∞),0<y≤552y∈/R,y∈(−5195,0)∪(5195,+∞),y∈(−∞,0)∪(552,+∞)
Find the intersection
y∈/Ry=−105+3205y=10−5+3205∪y=−105+3205,y∈(−5195,0)∪(5195,+∞),0<y≤552y∈/R,y∈(−5195,0)∪(5195,+∞),y∈(−∞,0)∪(552,+∞)
Find the intersection
y∈/Ry=−105+3205y=10−5+3205y∈/R,y∈(−5195,0)∪(5195,+∞),y∈(−∞,0)∪(552,+∞)
Find the intersection
y∈/Ry=−105+3205y=10−5+3205y∈/R
Find the union
y=−105+3205∪y=10−5+3205
Rearrange the terms
{x=5y13y=−105+3205∪{x=5y13y=10−5+3205
Calculate
More Steps
Evaluate
{x=5y13y=−105+3205
Substitute the given value of y into the equation x=5y13
x=5(−105+3205)13
Calculate
x=705−3205
Calculate
{x=705−3205y=−105+3205
{x=705−3205y=−105+3205∪{x=5y13y=10−5+3205
Calculate
More Steps
Evaluate
{x=5y13y=10−5+3205
Substitute the given value of y into the equation x=5y13
x=5×10−5+320513
Calculate
x=705+3205
Calculate
{x=705+3205y=10−5+3205
{x=705−3205y=−105+3205∪{x=705+3205y=10−5+3205
Calculate
{x=705+3205y=10−5+3205∪{x=705−3205y=−105+3205
Check the solution
More Steps
Check the solution
{5×705+3205×10−5+3205=133×705+3205−10−5+3205=4×705+3205−1
Simplify
{13=131.740198=1.740198
Evaluate
true
{x=705+3205y=10−5+3205∪{x=705−3205y=−105+3205
Check the solution
More Steps
Check the solution
⎩⎨⎧5×705−3205×(−105+3205)=133×705−3205−(−105+3205)=4×705−3205−1
Simplify
{13=133.168769=3.168769
Evaluate
true
{x=705+3205y=10−5+3205∪{x=705−3205y=−105+3205
Solution
(x1,y1)=(705+3205,10−5+3205)(x2,y2)=(705−3205,−105+3205)
Show Solution
